Switch jobs before October after H1-B approval (currently on OPT STEM extension)

[lazyme]

I am currently on an OPT STEM extension that expires in 11/13. I just got approved for H1B and that begins in October later this year. I know that on STEM extension you can only switch jobs if the employer is e-verified. Now since I have H1B approval, can the new company also apply for an H1-B for me this year now that the H1-B quota for the year is already full? If yes and if the other company is e-verified too, can I make the job switch before October?

Thanks!