STEM OPT Transition to Cap-Exampt H1B

My STEM-OPT expired on July 14. I'm still in the US. My employer is preparing to file my H1B. It should be filed before the end of the 60-day grace period. I will work for on higher ed institution. So the H1B will be cap-exempt.

Will I be able to stay in the US when my grace period ends but the H1B is still pending?

My wife is also in the US on F1. So I could also file for a status change to F2 first.