With that caveat said:
I would four bet preflop - you have postion and don't want to give up control of the hand. Why are you not four-betting? Almost any hand I raise UTG I four-bet here headsup except I may not with AA depending on if I think I can manipulate villian postflop well.
You can't fold to the CR on the flop. This could just be a standard blind defense move. You are seeing monsters in the closet and you have 4 outs to likely nuts and possibly overcard outs for one flop bet getting good odds in position. Just call as you did seems reasonable.
I raise turn, but again I don't see how villian's range could possible be what you said. (I have been involved in similar hands against tight old rocks that I might just call down here, but again I suspect that is not really the read here.) Not raising the turn here is really bad. You say "I am likely behind?" What???
The river is a call only. But I would have bloated the pot with many more bets so it would be an easier decision for me anyway
