"Proper ID" is the key, I think. FL, as other states, allows a grace period for establishing full residency. You have a period of time to acquire a FL driver's license and to register and tag any vehicles with the local county court house. During that grace period, you could have a VA driver's license and vehicle tag, but have a permanent, new FL address. Whenever I've moved state-to-state, I usually get that done as soon as possible. 20 years ago, when I moved from FL to GA, I recall I had several months to get my GA vehicle tags, but less time to change my driver's license. I could not get my GA firearms license until I had "established residency" in GA. That amounted to a GA driver's license, GA vehicle tags and a permanent address in GA.
You may have the logistics problem of getting a FL driver's license in order to acquire a resident carry license, which means there would be a period of time during which your non-resident carry license, as a VA resident, will not be valid, since you are carrying a FL driver's license and you would not yet have your FL resident carry license.