Semantic wrangling aside, here's how to answer the OP's question: Get a VA state-issued ID. If you want to do FTF transactions, buy from an FFL, or OC as stated, then you need to be able to pull out a state ID or Driver's License and show your address. This is the item the sellers of firearms or LEO's are going to ask to see to verify your residency for a firearms sale or to check on OC eligibility.
If you do not want to surrender your FL DL, then check into just taking the necessary papers (passport, rental agreement or utilities bills, etc.) and getting a VA ID (non DL). If VA allows dual licenses (a few states do; most don't), you can get both. FL issues a "FL only" DL for the snowbirds, so you could also get a VA DL and an FL restricted one.
"Residency" is where you live. It is not tied to who issued your DL, if you pay taxes, whether you are a welder or a student, etc. It may have a certain small time test to it, but that's all. If there are VA rights you want to exercise that are restricted to residents, you need a VA ID of some type - DL or other state-issued picture ID with your VA address on it.