Katharine McPhee is dating her former "Smash" director Michael Morris, TMZ has learned ... and they're even making out in public -- problem is, THEY'RE BOTH MARRIED TO OTHER PEOPLE.

Morris is married to actress Mary McCormack (who played Howard Stern's wife in "Private Parts") ... and they have 3 young daughters together, ages 2, 6 and 9.

29-year-old Katharine is married to 47-year-old producer Nick Cokas -- they have no children.

Sources close to Katharine and Nick tell TMZ ... the two have been separated for 6 months and are living apart. They were married in 2008.

It's unclear where Michael's relationship with Mary stands -- but it seems it's also on the ropes, because over the weekend he was sucking face with Katharine in broad daylight in L.A. like he didn't care who saw.

The two weren't shy with the PDA either -- holding hands and cuddling each other in Hollywood.