Defending against draws

[Openside]

I hate positions like these. Too often i just shove, which is bad. Lets assume villain is on a flush draw with an over card to our Qs. Therefore he has approximately 45% equity. How do i play this hand?
Note: this assumption is essential to my question. I do understand that it is other, non-draw hands in his range that allow us to value bet the flop profitably, though it is still pretty thin.

So, given the premise, if i shove he has equity to call. If i bet, how do i size my bet to extract value? i.e. pot odds (%) > villains equity (%). I can't.
Do we check behind and pray that the turn misses his range? If it misses then his equity is sufficiently reduced that we can make a value bet on the turn. Should the turn hit his range then its a cheap, easy fold. I hate the idea of giving him a free card.

Full Tilt No-Limit Hold'em, $0.05 BB (7 handed) - Full-Tilt Converter Tool from FlopTurnRiver.com

Button ($1.91)
SB ($4.25)
BB ($5)
UTG ($5.05)
MP1 ($5)
Hero (MP2) ($10.35)
CO ($4.81)

Preflop: Hero is MP2 with Q, Q
UTG bets $0.20, 1 fold, Hero calls $0.20, 2 folds, SB calls $0.18, 1 fold

Flop: ($0.65) 6, 3, 7 (3 players)
SB checks, UTG checks, Hero bets $0.65