You can travel to PR with your driver license and no passport required. But you must show your H1B approval notice and new I94 (also pos*****ily your past documents such as old I20 and old visas) at USVI upon departure to US mainland or PR. There is no immigration check upon arrival at USVI. BVI requires British visa.
But your flight cannot land anywhere outside US territory (50 states, PR) before you reach PR or USVI. Otherwise you must have a valid H visa to enter PR or USVI. So, take a direct flight to PR and in San Juan there are dozens of 30-minutes flights to St. Thomas.