Most universities approved by Department of State to issue a DS 2019 form will sponsor you for a J-1 visa. As Saurabh said, you can study on an H-4 but you cannot be paid a postdoctoral fellowship on that visa. F-1 is not a work visa and your fellowship salary is technically not a scholarship. Your J-1 will have a longer tenure (7 years plus) than an H-1. If your husband's H-1 contract ends for some reason, he can transition to a dependent J-2 with an EAD that has nationwide any employer validity as long as you are in status. If you are in biomedical sciences, you and your husband can transition directly from J-1 to a green card in the self sponsored EB-1 National Interest category provided you have publications.
All the best
Dr. Sandeep Shankar, Colorado Heights University, Denver, CO 80223, USA. www.chu.edu; email@example.com