That doesn't sound normal at all, but it does sound like you might need a new nest box design, and maybe better breeding does. Rabbits are very cold-adapted animals and should be able to have and care for babies in fairly cold conditions, certainly at temperatures you'd encounter in Georgia.
Here in south-central Alaska our rabbits routinely have successful litters at temps between 15-30 degrees F, sometimes as cold as 4-6 degrees above zero. (It does seem that having rabbits due to kindle is a sure way to produce a cold snap!
) If the mother does what she's supposed to, the kits do fine.
A few years ago I inherited some metal under-box heaters, but I have never actually used them. After a greenhouse fire due to a malfunctioning heat lamp many years ago, I have steered clear of electrical heat in my barn. The other issue is that making the nest box warmer than the ambient temperature can encourage the doe to sit in the box herself, which usually is a bad thing - see below.
Normally, mother rabbits should build a nest and line it with fur sometime before or during kindling; they will often continue pulling fur for several days afterwards. They will deliver the babies in the nest, clean them (sometimes some of the first-born kits nurse while she is tending to later-born kits), then leave them. Some does reach back in and cover the kits, and the best nest-builders even push up a huge wall of nesting material to conceal the kits. But this is not really necessary, as it is the job of the kits to keep themselves covered (or uncovered if it gets hot). They do this naturally by their digging and burrowing around and under each other and sharing the warmth while they push up a dome of super-insulating fur. If there is enough fur in the box, and enough kits (minimum seems to be 3-4), they will keep themselves toasty warm.
One thing that can go wrong is when young does, especially first-timers, don't build a nest or pull enough fur in time. Then the kits freeze due to lack of insulation to keep their shared body heat in the nest bowl. I usually keep a paper bag of extra fur on hand to put on top of a new litter that isn't covered with a copious amount of fur. I avoid ziploc bags for this, as fur in a plastic bag that will often create condensation and grow moldy, while paper keeps the fur clean but also allows it to air out.
Another thing that can go wrong is when a doe is too curious and repeatedly uncovers the kits. I don't know why they do this but it does occasionally happen. The kits can't survive too many repeated exposures.
The mother will usually go in once or twice a day, uncover the kits to feed them, then quickly leave the kits to re-cover themselves; most does ignore the nest for the rest of the day. Of course there is variation in behavior among individuals, but many or even most does do not brood the young or really make any attempt to keep them warm other than providing fur. You can imagine that in the wild this behavior would be quite adaptive, since the mother rabbit's presence in or around the nest would alert predators to the presence of the nest itself.
Knowing this is useful to the breeder who wants to "shelve" the litter, meaning to pull the nest box out and keep it on a shelf either in the barn or in the house or garage. Taking the box away from the doe when she's not feeding the bunnies can help keep bunnies from popping out; it also prevents the doe from uncovering them, or sitting in the box, or any of the other unhelpful things does sometimes do. When I shelve a litter, I cover the entrance to the nest box with a towel to prevent possible popping-out; it's not a heavy door but it keeps newborns from ejecting by accident. I usually take the box to the mother twice a day, morning and evening, unless I already know her habits (some does feed once a day, some twice). She'll hop right in if she needs to feed them. It takes a few minutes, then she hops back out, and I take the box back to the house. Our family actually really enjoys the process, and the kits get a lot of attention from day one, and end up super tame and friendly.
In fact, females that like to sit in the nest box when they're not feeding the kits cause the most problems for me. Not only do they soil the nest box and even worse, make it wet, but when the kits know she's there, some will crawl forward out of the warm nest bowl looking for lunch. If the doe is not ready to feed them, she can jump around trying to avoid them, possibly smashing them, or leap out of the box, possibly dragging some kits with her. Even if she doesn't drag any out, the litter is then scattered around in the box, and very young kits can sometimes get chilled before they can find the rest of the litter in nest bowl again. I wonder if this might be happening to your rabbits.
When I have a box-sitter, I usually give her a very small nest box, so that it is not really comfortable for her to sit in it. It's amazing how small a box they will build their nests in - some of my boxes are barely bigger than the rabbits themselves. They take care of their kits just fine, but don't spend any extra time in the box. I also give those does something else to sit on, in case they're in the box because they're trying to get off the wire.
Another trick that has helped us prevent pop-outs is making the nest box floor slope toward the back, so that gravity keeps the kits in the back of the box. We've done this several ways: actually building the nest box with a sloping floor; propping the box up in front (I don't really like this approach because if the kits still manage to get out, they sometimes get stuck underneath the box); or tilting the whole cage slightly. My cages are hung, so it's easy to adjust the angle.
One more approach to preventing pop-outs is to build a lip along the entrance of the box that extends inward, and acts to kind of scrape off kits as the mother is leaving. They are kind of a pain to build, and I haven't found these to be particularly effective, but some people really like them.
I don't have too many problems anymore, but I'm currently thinking about trying a below-cage nest box. It's a wire-bottomed box that hangs down under the cage. Kits can't fall out, they're less likely to be able to hang onto the doe as she leaves, if they do somehow get out they can roll back in, and the does don't seem inclined to sit in them. You do have to cut a hole in the cage floor, which is why I haven't done it yet, but it might worth trying.
Finally, I think that part of why I don't have many problems anymore is that I don't generally keep problem does. Mothering behaviors are definitely heritable, and I want our rabbits to be as naturally healthy and low-maintenance as possible. Even my first-timers usually do just fine. If I have a doe that is a persistent box-sitter, I usually remove her from the breeding line-up, too. I know that many beginning breeders simply have the rabbits they can find and afford, and don't have the luxury of picking the best-mothering does, but it is a good idea to keep it in mind for the future: in most characteristics, you get what you breed.
Hopefully some of this will be helpful to you. You should definitely be able to raise rabbits in Georgia's fall weather. Other folks on this forum will probably have some helpful thoughts and ideas as well.
