Negotiating an offer, is it too late? (update)

It is the first offer in 4 months and I do not want to mess it up. Thank you in advance. I have a competing interview next week. Did it backfire?

I thanked them and said I was excited but hoped we could talk about the pay. They came back with a bit more.

Thanks for writing this out. I went through something close to it 4 months ago, and it settled down once I kept a simple log of every call and form.