How to Negotiate Offer / When to Give Two Weeks?

Hey all -- recently received a verbal offer from a nichey Corp. Dev. / Strategy role in an industry I'm super passionate about (TMT-adjacent). 

The offer came in at the lower end of my comp range (i.e., I wouldn't say no, but was hoping for a bit more), so was wondering how to go about negotiating up another $10k? It's already a pretty serious pay cut from my current role but was hoping to keep the base salary essentially flat and that extra $10k would do it.

I haven't received an offer yet, so was thinking of having this discussion with HR when I receive that (rather than my new manager, who has been the main point of contact). Has anyone been in this situation before and has relevant experience? I don't want to lose the offer by negotiating for a small amount but I also want to set a fair base if I'm playing the longevity game.

I also want to maximize time off in between gigs, but wasn't planning on giving my two weeks until I sign the actual letter. I assume that's the way to go here but if anyone has other feedback that would also be helpful.