(1) It is not necessary to change your status from H4 to F1. You can enjob the residential tutition. Tell your international student office, H4 can study without changing F1.

(2) If your school requires you to change your status, you don't need to leave the US and get a visa. The procedure is pretty easy. File a I-539 to USCIS, and following the above document list. It takes about 6 months (depend upon your service center).

Finally, I would suggest you keep your H4 status. Save time and safer.