10/20 Limit HE, 10 seat Hand - could you have gotten away from it?

Don't know how to use the hand converter yet, but here goes...
I'm in Big Blind with Q10o, 1 early limper. Then it gets folded to the CutOff who raises. We both call.

Flop comes: 10 7 3 rainbow.

I bet, early player folds, CO raises me. I call.

Turn: Q.

I check (with the intention of Check-raising), CO checks behind.

River: 4.

I bet and CO raises. I re-raise/3-bet, CO caps. I call, and he flips over QQ.

Did I mis-play? Should I have 3 bet him or just flat called on the river?
Should I have folded to the cap?

I put him on an Over-pair on the flop...If he had a set of 10s, he probably would have
waited until the Turn to raise me. When he checked behind on the Turn, it seemed to me
to scream AK (maybe JJ or 99) or something...Thanks,