Boring/standard hand: Overpair versus poss. flush

[stigmata]

Hi,

My first NL post so apologies if this is a common/boring situation. FWIW, Iv'e just started playing some NL and i'm very inexperienced. Pokerstars $3/6 game. Villain has been sitting out since I arrived, and posts BB+dead in the CO. Therefore I have no read. We both have around the buy-in each ($600).

I get dealt JJ UTG. I raise 4xBB. The CO (poster) calls, and it's HU.

Flop: Tc 9c 2h
I bet 2/3 pot. CO thinks for a few seconds and calls.

Turn: 4c
I don't have the Jack of clubs.
I bet 1/3 pot. Although the flush hit, I'm thinking he could also have something like AcTx or QcJx etc. Villain quickly calls.

River another offsuit rag. I check and call a half pot bet.

Comments an all streets appreciated, but I feel like I mangled the turn and the river. I guess I'm used to paying off in limit games. With some sort of read I would call the river against an over-aggro chump, and fold against tighter players. Is a river fold standard against an unkown player?