Pudge told me he wanted to move so naturally I said "Don't go!"

He then proceeded to tell me that he wasn't sure he could get out of this one and that maybe if he asked SUPER nicely he could stay.

How can I stop him from going?

talk to him again if he's still asking about moving then choose don't go, i think he will say he wants to stay

That's how I remember getting villagers to stay when they said they weren't sure they could get out of it. For the most part it nearly always worked.