KQ OOP - Playing a draw

[Tosspot]

62 hands on the button and he's running 20/13. No real reads.

Ordinarily i would cbet this flop but I was paying full attention accidental checked instead. When the villain bet I consider raising but i really dont want to be 3bet. How do you play the turn? Would you ever lead here? Would you ever shove over his bet?

Thanks

Full Tilt Poker, $0.25/$0.50 NL Hold'em Cash Game, 6 Players
LeggoPoker.com - Hand History Converter

CO: $50
BTN: $51.55
SB: $50
BB: $133.90
UTG: $32.95
Hero (MP): $56.95

Pre-Flop: Q K dealt to Hero (MP)
UTG calls $0.50, Hero raises to $2, CO folds, BTN calls $2, 2 folds, UTG calls $1.50

Flop: ($6.75) T 2 J (3 Players)
UTG checks, Hero checks, BTN bets $5, UTG folds, Hero calls $5

Turn: ($16.75) 3 (2 Players)
Hero checks, BTN bets $11, Hero....

[esswun]

fold.
you have horrible implied odds, next time "pay attention" and bet the flop imo

[Poker Jon]

Tough spot, but I think as played, you have to ship raise it on the turn

[Nem]

[noidea]

you could donk turn, jam river.
if he doesnt jam that turn on the drawiest board in history when you donk he should never have a big hand.
with the right image he folds his 1 pair on the river.