There is an undercurrent of tension abroad in the otherwise sublimely becalmed Dorset town of Bridport. It emanates not from me, nor from David Tennant, and it emanates only subliminally, at some sub-woofer level perceptible to moles alone, from the smiley woman in the corner.

She is the ITV press officer. She is here to make sure that David and I behave ourselves with regard to plot giveaways, to anything giveaways. As it happens, it is only she who, with blithe, refreshing honesty, slightly gives the game away. I ask Mr Tennant how surprised he was by the roaring success of Broadchurch, the TV talking point of 2013, and she interrupts to gush – “Oh yes, it was our biggest hit for a long time!” David Tennant turns to her, his face gaping with laughter. “Surely you can’t say that! What about Downton?”

Series one of Broadchurch achieved near-universal praise. It was the best television series of 2013, according to Radio Times; Entertainment Weekly called it “a bona-fide national obsession”; it averaged 9.2 million viewers per episode, making it ITV’s highest-rated weekday drama series since 2004. A peak of 9.3 million viewers watched as the murderer was revealed. It was nominated for three Baftas (including a successful Olivia Colman for best actress) and won a coveted Peabody. Hence ITV’s efforts to let the BBC know all the drama gongs are not theirs by right, and hence also the “broadcaster control” which would have done immense credit to a paranoid Albanian border post in 1958.

But – sigh – it makes it all just so damned hard to write about. David, this splendidly likable Mr Tennant, also sighs. “Series two is absolutely propelled by series one. But it doesn’t have the same kind of crime-procedural structure – oh God, Euan, it’s so hard to talk about without giving anything away. Right. What I can say, is… hmm. What I can say is…” He slows down, finally.

Sands of time: with co-star Olivia Colman and other cast members of Broadchurch, in Dorset. Photograph: Patrick Redmond

“Broadchurch. Why Broadchurch, again? It was never the intention to do it more than once. It was always written as a one-off. And we actors were all contracted as a one-off: there was no sense it was going to go again. We were taken mildly by surprise when it became a national talking point like that.”

Parallels were drawn, repeatedly, to Scando-drama, specifically The Killing and Borgen, which concentrate as they all did on the generalised godawful fallout from even one killing, rather than a plethora of bodies. “I’m sure that’s part of it – the veracity of the repercussions. You can see ghastly murders on TV any night of the week – and they can become MacGuffins rather than human events. What Chris [Chibnall, the writer] did was show the impact that something like that would have, and it didn’t get neatly wound up in one episode or even one series. Even when you do find out the truth, there’s no reset button: there’s just the sense that lives are destroyed, communities changed for ever.”

So, with Broadchurch 2, can we expect more of the same? “No, actually, I don’t think you can.” I didn’t, I stress, mean it in a bad way – “No no, it wasn’t taken in a bad way. But it’s not the same at all. The notion that it might start with another body on another beach seemed preposterous if not a little cheapening of what went before. It couldn’t be Midsomer. Not that there’s anything, um, wrong with Midsomer, but it’s a different type of show.” But there’s a body, yes? A death? He speed-glances the PR’s way. She lurches up from the corner with cheerful gloom. “Aaaah. He can’t say.”

Mr Tennant boggles an eye or two. “Wow. They’re really scared! Is that why you’re sitting here? In case I say the wrong thing?”

“Kind of, yes.”

“I won’t give anything away. I did Doctor Who for four years – I know how to do this stuff!”

We can stop talking about it, I suggest, and he visibly relaxes. Shorn of the need to stilt-walk on eggshells round putative plots, he and I enjoy a thoroughly liberating trip down to the beach, talking, in the very lovely English coastal sunshine, of Scotland and, of all things, its church. I’d read that David’s father was the Moderator of the General Assembly of the Church of Scotland, and murmur that the minister who married me now holds the same post: he knows of John Chalmers, and is aware of his son JJ, who was grievously wounded in Afghanistan, and so the talk veers wildly between war, and the Proclaimers – he’s a huge fan, and walked down the aisle to Life With You – and identity, and the Jurassic Coast, and the cast party the previous night.

‘I enjoy the language of Shakespeare, I enjoy wrestling with it’: playing Richard II, with Michael Pennington as John of Gaunt. Photograph: Tristram Kenton

I have soon to surrender him again beach-side to the camera’s ministrations – he does hate being forced into unease and lack of naturalism, whether mental or physical, and asks just half-jokily of our photographer, “Why do you hate me, Alex? I honestly don’t think I’m going to able to do this, from a shaming point of view. I’m not, to be honest, very good when I’m not pretending. Sorry.” But, away from awkwardness, I realise this is a very wise and happy man indeed – just give him a pair of torn jeans and a sweatshirt and he is delighted, hugging his bony knees and chewing on wind-lashed hair, to gas away about pretty much anything, and fast and fluent and funny – anything as long as it doesn’t involve complex broadcasting protocol, or monkeying about for a lens. Or his personal life.

He has recently, for the record, had another son with his wife Georgia Moffat, also an actor and whose father, in one of those loopy coincidences you might think profound but which turns out to be worthy only of a second’s raised eyebrow, is the former Doctor Who Peter Davison. But he wants to talk instead about different, interesting subjects, such as Shakespeare, and the parts he’d love still to play.

“Iago is a bit of a juicy one. Malvolio I did on the radio and that gave me a taste for doing it on stage. Angelo in Measure for Measure – I think that’s a fantastic, brilliantly complicated play. Also, I’d love to do a bit of Arthur Miller, that whole era of American Realism, Eugene O’Neill through to Tennessee Williams. Then, of course, there’s all the parts you can’t imagine because they haven’t been written yet, which are the most exciting.”

Gregory Doran, artistic director of the RSC, has said, of his Shakespeare: “Somehow [with David] the lines don’t sounds like 400-year-old lines.” How does he manage that? “I’m delighted Greg thinks that but… oh God, how do I answer that? I don’t know. But I enjoy the language of it, I enjoy wrestling with it – I think, when you first sit down with Shakespeare it feels like it’s driving you. And I enjoy the process of getting to the stage where you’re in control of those words. Because they’re such extraordinary words – yeees, they can be a little difficult to master ’cos they’re old and sometimes archaic – but when you feel you’ve got the wheel, and can jazz with it a little, to be in control of that vocabulary, that poetry, is wonderful.”

You’ve said yourself, relatively recently, that “theatre still feels like the day job.” Is that changing? A nod. “Certainly the balance has changed. At the moment. I dare say it’ll swing back again, because these things do, over the course of hopefully a longish career. It’s a funny thing, theatre. Every time I do a play, I say to myself the first night, why the fuck am I putting myself through this balls-aching torture?”

Need for applause?

He laughs. “Thanks. That sounds very needy. And probably true. But there is something – I do feel compelled to do it, it does still feel like the ‘job that I do’. I probably do feel safer in theatre, I still feel I know what that world is more. But, yes, anyone can fail. I suppose what’s difficult in theatre is, if you’re not making it work you still have to churn it out for eight performances a week for the length of the run which can be a little soul destroying.

Time traveller: as Doctor Who with wife Georgia Moffett when they appeared together in the series in 2008. Photograph: Adrian Rogers/BBC

“But I’m hopefully doing a film in the new year, though films being films you don’t want to rely on them till day one, and this TV stuff, and I love it all. If a script comes in, captures my imagination at all – I think, well I don’t want anybody else to do that. There’s a sense that if you’re fortunate enough to be in this profession, and being offered work, then it would be churlish to turn it down. And, yes, there’s a certain amount of Scots Calvinist guilt in all that – in any freelance lifestyle you can’t rely on it past the next contract end.”

Does he, I ask, feel guilty often?

“Oh please, Euan! Everything. All the time! Absolutely. Guilt is my engine in life. Working too hard, not working hard enough, both extremes at the same time. I’ve actually learned to enjoy it: as an engine I think it can be harnessed for the greater good, as long as you can keep it in check. It doesn’t overpower me as much as it used to, though that’s probably just old age. I truly don’t mind a bit of guilt, I think it can be quite useful.”

He is predictably mumbly when I move on to sex-symbol status: regularly voted the sexiest Doctor Who ever. “I don’t think I am a leading man. I’ve had some parts which have been defined as leading parts but… hmm. Err. I think I’m probably Richard II rather than Henry V, that’s how I would see myself – and I’m rather happier there. I think square-jawed directness is always less interesting than slightly left-field complexity.”

We’d been talking earlier about the TV series The West Wing – he is possibly Britain’s most demented fan, and he says, today: “I think there should be many more political dramas on TV: The West Wing managed that on such a scale, gloriously.” Tennant’s acting history tells its own political tale: he was a well-regarded member of Scotland’s 7:84 theatre company, named after the 1966 statistic (now worse both nationally and globally) that 7% of the UK population owned 84% of the wealth, but here on the beach, as we discuss Scotland’s referendum, he won’t be drawn: “I just don’t think I should have a say, I’ve been out of the country for years. And look at the hellfire that descended on Andy Murray and poor JK Rowling – both sides.” No, today the American drama is purely being used an illustration of his antipathy to phwarr status.

He draws a comparison between that show’s Rob Lowe’s clean-cut dullard and the actor who plays Josh Lyman, the mesmerising deputy chief of staff. “I think, in all honestly, if I could be Bradley Whitford I would be very, very happy.”

He becomes almost drawlingly dreamy, rolling his “r”s as he leans against the warm oolite cliffs of this Jurassic Coast, until rudely interrupted by me, asking whether there’s talk of a Broadchurch 3. There is a timely rescue from the PR, who has been struggling up the sands towards us. “Who knows?” she beams. Tennant turns, grinning. “Apparently that’s the answer I’m meant to give. I’m so glad you’re here.” He’s so warm, tolerant, benevolent, chuckling, towards this charming emissary from corporate control, and such a good actor that, for a moment, I almost believe him.

David Tennant on the beach at West Bay in Dorset Photograph: Alex Lake for the Observer

The seconds series of Broadchurch is on ITV in January