I think you have to shove the turn / river given how you played and hope that scares them off, but given how it played out I think they have JJ+ so you're stuck.
A better line is probably check raise / check call the flop. Your bet on the flop doesn't do much as it's pretty clear neither of you have a 6 form the pre-flop action, and if you did have 66/77/67s, you wouldn't be leading it, you'd be looking to slow play.
The check raise is the only way you're scaring villain off their hand given they probably have a better overpair or AK and are happy to float you on the basis they know you don't show up here often with a 6 or 7. They may also float you if you've exhibited this tendency before (leading OOP on flop and then shutting down on future streets if called).
Given your line - if you weren't prepared to lead the turn betting, you have to fold. You can't check call here as you're letting them set the price to see the next card (possibly for free if they check behind) and there's a tonne of rivers that kill you (i.e. any broadway). As it happened, even a low river did for you so should have folded the turn - what river would you have played? Only a T? If so the turn is an insta-fold as you have 4% to hit and you're calling a half pot bet
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