Okay, I've been working a while with many different fields of music... I certainly wouldn't say that vocals are my specialty; however, rather than doing what every other answer suggests (having you use a program that "trains" your intonation to become more in tune), I think it'd be a lot more effective and genuine if you were to just simply work on the concept of developing your abilities to detect and utilize consonance (the concept of a note being complementary to another sounding note). So, in the case of developing your ability to become more "accurate" with intonation, try to gain an appreciation for developing a skill of harmonic consonance more so than singing in unison with a note. By doing so, I thoroughly belive that your ear will be more adaptive and trained to handle singing IN UNISON when you need to.

In the case of a sustained C, for example, other notes you should try singing to develop a sense of harmonic consonance would be E, F, G, and A. Hope this helps. If not, try someone else's answer!^^

Best of luck!~ ;D

EDIT: To emphasize, this is important for developing a sense of how that OTHER note fits harmonically (and melodically) into a song/chord/section/anything.

SECOND EDIT: I seemed to have excitedly overlooked a very important aspect of what you were saying in your initial comment, that is, you can HEAR the dissonance (which is good), but focusing on achieving consonance over (the inverse) avoiding dissonance (by my personal philosophy) is key.

For example, some people play chess with a mindset of not getting into a certain situation or not losing (rather than playing with a mindset of winning or trying to force the other player into a certain situation). So, in this case, I feel that, rather than avoiding dissonance, you should try to gravitate your efforts in finding consonance... To find consonance, you don't HAVE TO sing in unison with the note in question.

And, then again, a lot of people are good at chess by simultaneously working on both approaches; I'm sure the same can be said about vocal intonation. I dunno, I guess I've over-emphasized what I'm trying to say, but really, I feel there are usually multiple solutions to solving a problem.