H1b/i 94 near expiry help!

[swapstx]

Hi I work with a university. My current job and I-94 expire on september 30th this year.

If I get a new job offer, what does the new employer process, a new h1b or a h1b transfer?

Can I start working for a new employer once the petition is submitted or do I have to wait for an approval from USIS? If I have to wait, then can I stay in this country or do I have to leave and re-enter?

Thanks for any help in this regard.