How do I play these hands? :/ Table is loose.

Hand 1:

Pre-flop: (10 players) Hero is SB with
5 folds, MP3 calls, CO calls, Button raises, Hero 3-bets, BB folds, MP3 calls, CO calls, Button calls.

Flop: (13SB, 4 players)
Hero bets, MP3 calls, CO folds, Button raises, Hero calls, MP3 3-bets, Button calls, Hero calls.

Turn: (11BB, 3 players)
Hero checks, MP3 bets, Button calls, Hero raises, MP3 3-bets, Button calls, Hero caps, MP3 calls, Button calls.

River: (23BB, 3 players)
Hero bets, MP3 raises, Button calls, Hero calls.

Results:
Final pot: 29BB
MP3 shows Ah Ad
Button doesn't show Jd Ac
Hero doesn't show Js Jc

Same session:
Pre-flop: (10 players) Hero is MP1 with
UTG calls, 2 folds, Hero raises, 3 folds, Button calls, 2 folds, UTG calls.

Flop: (7.4SB, 3 players)
UTG checks, Hero bets, Button calls, UTG calls.

Turn: (5.2BB, 3 players)
UTG checks, Hero bets, Button calls, UTG folds.

River: (7.2BB, 2 players)
Hero checks, Button bets, Hero calls.

Results:
Final pot: 9.2BB
Button shows Tc Ac
Hero doesn't show Qc Qd

Another one:

Pre-flop: (10 players) Hero is UTG+1 with
UTG folds, Hero raises, 3 folds, MP3 calls, 2 folds, SB calls, BB folds.

Flop: (7SB, 3 players)
SB checks, Hero bets, MP3 calls, SB folds.

Turn: (4.5BB, 2 players)
Hero bets, MP3 calls.

River: (6.5BB, 2 players)
Hero bets, MP3 raises, Hero calls.

Results:
Final pot: 10.5BB
MP3 shows 9s 9h
Hero doesn't show Ac As

I think bet/fold >>>>>check/call>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>bet and call a raise which is what OP did.
No one bluff raises this river.
So if you can't fold to a raise, don't bet in the first place.

But if you CAN fold to a raise (and we should strive to be able to in this hand):
Here are the salient points:
I think that QQ/KK/TT/Ac9 all call here and are all plausible hands (Ac5 is a stretch to call preflop). So there are some hands that call a bet.
The problem with check/caling is that if villian had a set (now river boat), made flush, or a J we can assume he always bets river if checked to, and we have already said we have to call if we check.
So we really don't "save a bet when we are beat" as bigwheel said, as any player bets any of those hands when checked to and those are the only hands that beat us anyway and we are calling.

So really this is pretty simple given the above statements, there are only two variables left:
If[ times villian has QQ/KK/TT/Ac9 and calls] is > [times villian bluffs river with unimproved Acx] = then betting is +EV.
I don't think villian bluffs much here so inducing a bluff is not a big %.
So therefore I bet/fold.

Tell me if I am missing something.

I think I would have a hard time folding to a river raise after my bet in this situation, even if I am 80% sure that I was beat...

I guess that is why I advocated the Check Call...

Leakage? Probably, but...

If I was 4 tableing, and this hand played out the exact same way at 2/4 where I am currently playing, I think I would call the raise just about every time. You only have to be right every once in a great while for you call to pay off.

Maybe I am playing too many tables not to be able to play this hand as you guys suggest. I guess I would rather be a little more passive and give up an extra bet rather than have to decide if folding to a river raise in a 12BB pot is the right thing to do in less than 20 seconds.

Something that I need to spend some more time looking at, or maybe you guys need to try and beat some sense into me...

I am kind of taking over this thread a bit, but I think this is still relevant to the OP original question.