$50 nl, flopped set on broadway board, line check

$50 nl, bb=$.50

Relevant stacks
Villain $103
Hero $57

No reads on villain except that he has doubled up on the maximum buyin and I have not noted any donktastic plays.

Dealt to Hero JcJs

PREFLOP: Villain limps, Hero raises to $2.50, one caller, villain calls.

FLOP ($8 in pot) Kc Jh 10c
Villain checks, hero bets $4, third player folds, villain calls.

TURN ($16 pot) 3h
Villain checks, Hero bets $12, Villain calls

River ($42 in pot) Ad
Villain pushes relatively quickly, Hero pukes, wishes he could find a call and be right, pukes again, and folds.

I have been mainly a Sit and go player and am trying to transition to some more cash game play. I feel pretty good about the preflop raise, but comments appreciated. On the flop, what is your play? I half potted because I wanted it to appear like a continuation bet and hoped top pair or two pair might come over the top. Also wanted to make draws pay but also wanted to wait for turn so I could get away if a terrible card like the 9c or Ac showed up. On the turn, should I have full potted instead of 3/4? I still wanted to extract value and not overbet push. Finally, on river I just couldn't find a call without having actually seen the villain either (a) make a smaller value bet when hitting an obvious straight, or (b) bluffing all in on an obvious scare card. Without this read, I feel like it is unprofitable because too often he is check calling with the Q (and maybe a K or 10 for a pair) and pushing when his straight hits. Absent a specific read, can anyone find a call here?

Comments on any or all streets appreciated.