This is so funny as I also have one missing a large portion of his tail since last fall and I call him stumpy too. We have him so tame he will also come and sit next to us also and I have tiny superworms he will jump into my hand and eat,
As to your question, yes they do their own version of hibernation called brumation. Instead of using stored fats like mammals do, reptiles use glycogen that is release slowly to meet their body's needs so they can survive through a winter. The do need to hide deep enough into the moist soil or another location so they do not freeze, as that will kill them. They usually dig into the ground and another place you may find them is around an outside ac/heat unit that is blowing warm air. They also are very adept at getting into any cracks in the eaves, roof or under houses if they can and surviving there where warm airs come from the home.
Just for general information, the small brown ones in Florida are called Cuban Brown anole and are not native to Florida. The green ones are native.
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My assistance should not be a replacement for competent hands on veterinarian care. Your vet should always be contacted when your pet's health is possibly compromised