Here is your predicament:
Any bet commits you to the pot with your pair of sixes. You need to hit one of 15 cards to give yourself a chance of winning the pot (but even then, some of them might not be good for you).
The flop bet I like, though I think I bet smaller, around $2 (I shove if raised of course) because I want to keep opponents in with my nice draw and the sidepot is completely dry at this point.
However as played, your only play is to bet full pot again. You can consider checking if your opponent is passive and wont bet himself, however if you do this your hand is pretty much face up. The turn card has changed nothing, so if you bet full pot on this flop it makes sense to jam the turn as well.
But if opponent is passive then Check/fold to a full pot bet as he wont bluff you, the sidepot isn't that big, so he may just check behind if he can't beat a big pair.
Anything less than full pot and you can consider a call for value with your outs.