We have been leaving in January for years and have never found it to be a problem. We make sure the roof is swept mostly clean before we leave. We leave our trailer as empty as possible, meaning no food, water, or beverages. No since in carrying any extra weight for 1200 miles as we have found they sell everything we need in Florida, plus some things we don't need.:)
We also Motel it the first two nights as we find it so much more enjoyable to relax without having to set up in a cold trailer. After another 7 hour third day of traveling we make it to the north end of Florida where we have made a reservation at a favorite campground for the night. Then on to our destination from there.
Like others have said, you do have to watch the weather for any storms, but other than snow or ice storms the express ways are always clear.