H1b transfer without I797A

Hi,
My H1B was supposed to expire on 09/01. My present company (Company A) filed for extension. I got an offer from Company B. Who told me until unless my extension is approved they cant file for H1B transfer.
Company B suggested to apply for premium processing, which I did with my present lawyers from Company A.
My visa petition now has been approved, but I havent received my I797A. It has been 3 weeks since approval but i797A hasn’t come yet.
Can company B file for H1B transfer with just I797C, along with USCIS Case Status Screenshot stating that my petition was approved? At this point I dont know when my I797A would come and I dont know if the new employer Company B would wait till then.

Key Facts:
I came on F1 visa, which got transferred to H1B. I have not left US since coming here, hence I dont have H1B stamped on my passport. Even my I94 Entry as F1.

Can anyone let me know what are my options and can Company B transfer H1b without I797A?

Thanks.

Yes, this should be doable. Discuss with your lawyer.