Did I misplay my set?

This hand I think I could have gotten away from cheaper? Or did I make the right play?

No hand history, but Ill recap:

Short handed 100NL 5 max
People/position/stacks for this hand:
Villan: Button (~$150), Hero: SB(~$130), Shortstack: UTG (~$30)

Reads: Villan is a lagg, likes to push people off hands. Id seen him make a few marginal plays though.

Preflop:
Cards delt to me: 4d 4s

Preflop action:
Shorstack calls $1, (UTG+1 folds, c/o folds) Villan Raises to $3, Hero calls $2.50, Shortstack calls $2.

Flop: [4c 8d Kd]
Hero Bets $7, Shortstack Calls $7, Villan calls $7

Turn: [4c 8d Kd] [6h]
Hero Bets $15, Shorstack calls $15, Villan Raises to $30, Hero Raises to $60, Shortstack calls all in for $5, Villan Pushes me all in for aprox. another $60.

Can I Fold?

(In retrospect would have it been better to just call the guys raise on the turn and see what the river brought?)

If he has the straight then ugh what luck and what a play. (cant see that though)
If he has a set higher than yours then . I bet he has AK or something similar possibly even AA?.
I'd call with my set everyday here. If i have one out then thats just unlucky. If im ahead then great i'm all in and counting their outs.

As to calling the raise no way. I would think im ahead here so im pushing the turn for that reason if not also for the fact the flush draw is there. (His AK may be suited so tp+flush draw)?
Lots of possibilities but i'd guess im ahead.

Shove the rest of it in on the turn. Any hand that has me beat there I'm happy to pay off.

He flipped 75d, a well-hidden double draw off the flop which he easily got odds for after short stack called my flop bet.