The way I see it, you were being underpaid :-) Otherwise they wouldn't offer you twice as much as you made before.

Back in the place I was before, when I said I was leaving, they didn't made me an offer; instead, they asked me to tell them how much I wanted to make and told me they would give me that figure. Of course, the fact that they had been telling me they couldn't pay me more for so long and then the sudden change made me even more decided about leaving than I was before...