I wouldn't worry about the higher set. The only thing I would worry about is the straight. You have 10 outs if he does have the straight. Not enough for the pot odds or anything, but just something to keep in mind.

I think he could have 'slow played' 2 pair on the flop. The overbet is kind of weird unless he made his gutshot straight draw on the turn. Anyways I think I would push the turn.

So to put him on the straight you would have to have a read that he calls too often -- called a pot sized bet on the flop with the inside straight draw only. And then a read that he only overbets with close to the nuts. A kind of loose-passive profile. Otherwise I second guess pushing the turn too much. There are enough 2 pair possibilities and I don't fear the overset here much at all.

edit: ok, i just realized it was a double gutshot so maybe there is some consideration for a fold here