$1000 NLHE Full Ring: Playing 66 out of position with 2 overcards

This was a live cash game with $5/$10 blinds and a $20 straddle. I don't usually play stakes this high and I wasn't comfortable bringing $1000 to the table, so I was playing this game short-stacked.

Hero (MP) has $600
Villain (BTN) has $3300
Hero is dealt 6h 6s

SB posts $5
BB posts $10
UTG posts $20
UTG+1 folds
UTG+2 folds
Hero (MP) raises to $60
HJ folds
CO folds
Villain (BTN) calls $60
SB folds
BB folds
UTG folds

$150 in pot
Flop shows Qs Kh 3c

Hero checks
Villain bets $80
Hero calls $80

$300 in pot
Turn shows 4d

Hero checks
Villain checks

$300 in pot
River shows 7h

Hero checks
Villain bets $180
Hero folds

Did I play this correctly? Should I have called the river bet in case he was betting on a busted draw like JT?

depending of the looseness of the table and us not being deepstacked i don't know if i even play that hand.

as played i cbet the flop to bluff out pairs like 22-TT and Ax - if i get a call i give up the hand

With only 30 bb stack,i dont like that check/call on the flop,or cbet or check/fold depending on the info you have about villain.

Typically playing baby pairs pre from MP and UTG with that flop, the advantage is we can still rep a strong hand and get folds if we cbet.

typically give up if called. Or we get a 6 on the turn.

Also just remember playing baby pairs is not the best in MP because our opponents in later streets are less likely to have very strong hands for us to get paid off, so setmining isn't always the best but cbet bluffing I'm okay with.

typically short stacks have a lot less fold equity and seem fishier.

Bet the flop as played. The strategy of check/calling oop to fold to any other bets won't be profitable overall. And yeah reconsider your preflop game a bit if you're going to deliberately buy short into a game you're not sure you have an edge in. If you have both short effective stack depth and a lack of fold equity post then it's really difficult to open 66 there.