Offer arrived, can I still ask for more? after 13 years as a project coordinator

I want to negotiate but I am afraid they will withdraw. I have read a lot of threads here and could not find my exact situation. I have a competing interview next week. What wording worked?

Most employers expect a short conversation. Mine did not even blink.

Keep everything in writing and save the confirmation numbers. It made the follow-up calls so much easier.