PokerStars No-Limit Hold'em, $0.02 BB (5 handed) Hand History Converter Tool from FlopTurnRiver.com (Format: FlopTurnRiver)

BB ($3.98)
UTG ($1)
MP ($2.59)
Button ($6.39)
Hero ($2.21)

Preflop: Hero is SB with Q, Q. UTG posts a blind of $0.02.
UTG (poster) checks, MP calls $0.02, 1 fold, Hero raises to $0.12, BB raises to $0.22, UTG folds, MP folds, Hero calls $0.10.

Flop: ($0.48) 7, 6, T (2 players)
Hero checks, BB checks.

Turn: ($0.48) T (2 players)
Hero bets $0.22, BB calls $0.22.

River: ($0.92) T (2 players)
Hero checks, BB bets $0.5, Hero calls $0.50.

Final Pot: $1.92

What's the best way to play these hands where you've got a big pair but someone else is suggesting they have a bigger one?

last time i was in this situation with QQ, i shoved all in and got called by AA, is the way i played this largely correct?

I bet more on the turn and I value bet the river. Did he really have KK here?

Bet more on the turn. Definitely bet the river. And don't be results-oriented!

Without reading the title of the thread which totally gives away the results, I would definitely bet the river and if I check, I'm CRAI here. Nothing about his post flop play says he has an overpair.