If I understand it correctly, Hero was OOP and Villain had position on Hero. Yes?

If that is the case, I c/c the river. You're getting very good odds to call, and he could have been calling down with a medium overpair just as often as with a flush draw. The king could be a scare card for him, but when you check he might decide to bluff you.

And the $10 on the river really doesn't make sense. That's less than 1/4 the pot. If you're going to bet that small, just check. As played, you have to call because you're getting great odds on your money. This situation just sucks, but I don't think you played it badly, except the river is a little wierd.