How much room is there after an offer? (update)

It is the first offer in 11 months and I do not want to mess it up. I have a competing interview next week. Did it backfire?

Ask about the range, not a number on its own. It felt more collaborative.

Thanks for writing this out. I went through something close to it 11 months ago, and it settled down once I kept a simple log of every call and form.