Timing of departure to Korea after resigning from H-1B

I’m working on H1B and just resigned, with my contract ending May 18th.

Generally, I understand there’s a grace period even after H1B is terminated, but my school’s HR said that only applies if you’re looking for a new job in the U.S. — since that’s not my case, they told me I need to leave before the 18th, and to consult an attorney for anything beyond that.

(Looking around this board, I’ve seen posts saying you have to leave immediately upon resigning from H1B, which is worrying.)

I’m planning to leave on 5/25 (a week after my contract ends) to pack up my things and so on — I’d like to know if I really can’t stay even that extra week.

I’d also like to know what action I’d need to take if I want to stay that extra week. (Would I need to file to change to B1 status in this case?)

Since I’ll need to come back to the U.S. about a month after leaving on 5/25 for a conference (I was planning to use ESTA for that), I’d like clarity on all of this.