Hopefully someone will be able to chime in before you go to give you better info. All of mine is coming from DeLoach's Dive Guide book for FL. It looks like the closest springs to Panama City are north, near Marianna. The list for that are is:
1. Merrit's Mill Pond/Blue Springs; Twin Caves; Shangri La Springs; Indian Washtub; Gator Spring.
2. Blue Hole
3. Bozell Springs
4. Gadsen Spring
5. Cypress Springs
6. Magnolia Springs
7. Vortex Springs
8. Ponce de Leon Springs
9. Morrison Springs
There's always the option of stopping near Tallahassee, depending on how your getting down there. Near Tallahassee, you've got:
1. Wakulla Springs
2. Wakulla River
3. Wacissa River Springs
I don't know anything about the drive times, or about each particular spring, but this should give you something to work with. Someone who lives in North Florida would probably be a better source of info.
Have fun and dive safe!