Move From an F1 Visa To an H1B Visa ( Can i stay if application is pending)?

I was recently offered a Job as a professor to start in the fall and they are willing to apply for an H1b visa (cap exempt), I am currently on opt which will expire before fall. However, due to the change in laws I can’t apply for premium processing for my H1b, can i start work in fall whilst i am waiting for my h1b, also if my application is pending I am allowed to stay in the country?

Which country are you planning to stay? You cannot apply for an H1B visa inside the US. You’ll have to go to a consulate outside the US. It’s better to apply for any visa in your home country as visa processing can take some time and since your passport will be collected.