Got a verbal job offer but no written offer yet — starting to panic

I got a verbal job offer last Monday (April 28) and was told HR would reach out. It's now Friday, May 1, and I haven't heard anything from HR or the hiring manager despite following up.

To make things more complicated, in my follow-up email, I also asked about my schedule, whether I could have flexibility on my start time, and mentioned I need May 22 off for my wife's graduation. I'm now worried that asking all of that before even getting the offer letter may have been too much and turned them off.

On top of that, my start date is May 18, and I still work at my current job. I need to give two weeks' notice by Monday, May 4, for everything to line up, but I can't do that without a written offer letter.

Has anyone been in this situation before? Is a week of silence after a verbal offer normal? Did I mess things up by asking too many questions too soon? And what would you do about the notice deadline?