Leasing a 10K car maybe a cheaper alternative to buying a 10K car. However have you considered simply driving your car down from up north? Let your wife follow you.
I understand we all have a budget and standards but $10K seems like a huge number for a simple grocery getter with A/C. I would think you could by a decent car for $5K.
Lastly before spending $10K for a car drive or ship yours down for the first year and see how you enjoy the overall experience before investing another $10K.
If you enjoy your new snow bird lot leave your current car in FL and by another car when you return to your stix n brix home up north.