I was in a hand where I thought based on my opponent's play, I was making the right decision to slow play a set vs. my opponent who is a TAG player, often folds when played aggressively against. For some reason, the way the hand went down, and the size of the pot, I made a terrible call for about $900 into a pot of roughly $560.

In a live $3-5 table, effective stacks are $1,000, my opponent has maybe $2,000, and I have about $1,300.

Preflop:
2 limpers, my opponent raises to $45 and it folds to me in the BB, and the two limpers fold. I call with 4-4. The pot is $98.

Flop:
Kh Qh 4x

I check to my opponent who continues for $45, and I call. The pot is now $188.

Turn:
Jc

I check to my opponent, and he continues to bet, this time $85. I think for some time before 3-betting to $185, assuming that if the river is not a heart, I should be good and lead out $300 for a call. After tanking for some time, my opponent calls.

River:
3c

For whatever reason, at the time, I deviated from my plan and checked, hoping to check-raise for maybe $400. My opponent went all in. After thinking for a while, I thought of the combos of Ace-x he could have, but didn't put him on KK, QQ, JJ, etc. since I assumed he would bet more with those hands, maybe 4-bet on the turn. There was no 4-bet pre-flop, so certainly A-10 could have been possible but I would not know. I did not think he would bet $45 with 9-10, so the only hand that would make sense would be A-10 or a bluff with 2 hearts, maybe A-x? Anyhow, I did not put him on A-10, and thought he semi-bluffed to the river, and I called with 4-4 and was disappointed to see that he showed A-10.

Analysis:
The fact that I even paid off about $900 after an initial investment of $280 is a pretty terrible decision without the nuts in that situation, or even second nuts with a straight. A ton of sets beat me as well. Did the hand begin to take a downturn at the flop, turn or river the way I played? Certainly in hindsight I think a check-raise would have been a better move on the board since it was pretty wet, and on the turn, if I used such a small sizing I most certainly should have been leading the river for value and if I was 3-bet I could easily fold. However, I put myself in a difficult decision, and assumed because I checked, my opponent went all-in to eliminate the possibility of me calling with 2-pair or a set, because he was in fact repping A-10. Does anyone have any thoughts to how I should have approached this hand, and how to correct my logic next time to include the fact that opponents could have flopped and turned a lot of bigger sets, and almost would never go all-in on the river with those hands? Or is it the fact that I decided to slow play, and get lost in the hand when the turn came? Should I always be folding the river in this spot? Thanks.