Incorrect preflop, and incorrect fold.
preflop: fold, your getting a bad price with a mid pair. if you play the button as you describe at limit hold em then you will be a big loser over time.
flop: fine.
after that you cannot fold a set in limit with the pot this sized. if you think he has a set, you call him down. you get shown AK here often enough, or even a big suited diamond draw with a pair (was the A on the board the ace of diamonds?), that you have a positive ev calling down here with a set.
at limit your supposed to lose a lot with set over set, at nl you need to be able to get away from bottom set.
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