I would apply to both and see who gets you in first. I think both have coordinators who work with patients to find housing. There are 2 military RV parks here and some stay at Flagmingo RV Park north of town. Those staying there like it. I pay approximately $100 week here at NAS JAX and I believe Flamingo is $600 month. I have heard housing is 1000 month and up but most seem to find places.
For Florida call (904)588-1800, or toll free (877)686-6009 that will be the switch board. Ask to speak to Katie Mahoney. She has housing information. No information on Bloomington but look them up on the net and call.