In the end, I'm glad I turned down the counteroffer and got out - I do think that my relationships would have been poisoned if I had stayed, and some core issues would not have been fixed. That said - I should not have taken the job I did. Chasing money was a rookie move, and I'm now more unhappy than I was at the previous company.
My advice is to think very realistically, perhaps pessimistically, about what life will be like at the new company. If you're not sure, ask questions. Don't take a new job because you want to get out - take a new job because you want out AND because the new job is freaking amazing. I'm now job hunting all over again, and it sucks.