Flop Nut Flush with an Ugly Turn

[donkey]

2/4 NL
just sat down, no reads.
I have 400, Villain has 400.

Villain (UTG+1) raises to 12. 4 callers. I call on the button with Ad6d. Blinds fold.

Flop is KdTd2d. Villain leads for 20. everyone folds to CO who goes all in for 80. I pretend like i'm thinking for a minute and finally smooth call. Villain thinks for a while also and eventually calls.

The turn is the ugly, ugly Ts. Villain checks, i check behind.

River is another ugly card , the As. Villain bets 80. I fold.

Was my flop play bad? should i bet the turn to find out if it's KKK or just AA? do i need to call this river? i figured since there's an all-in player, villains small bet is for value, not to bluff me out of a dry sidepot.

any comments appreciated.