Last January we snowbirded by driving from SE Michigan to west Texas. First night stop was at the Percival Springs Airport and RV Park(!!) in southern Illinois. This tiny park has electrical
hookups and showers. Glad we did this because it was windy and about 32 degrees with a couple of inches of snow and ice on the ground. The next stop was a municipal campground (with hookups and showers) in North Little Rock, Arkansas right on the river. At this point, didn't actually need hookups because the temps were well above freezing.
That's what we did. For the first night on the drive down temps are usually freezing. All we have is a Mr Heater Buddy which we prefer not to use if we can find a facility with electric on that first night.
So before we travel south, we spend a few hours checking Allstays and other info sources to find suitable stops with hookups for the first night or two. In addition we identify campgrounds/parks where we would like to stay at our destination. We copy down in a spreadsheet the name, address, phone, hookup info, etc of all facilities.
The alternative is to wing it and just stay at Walmarts, etc on the way down. We have done this but pretty uncomfortable in below freezing temps.
Also, on a previous trip south we got stuck in a campground in Kentucky when 6 inches of snow fell overnight. The campground was several feet below the road level... So now we try to stop at facilities that are close as possible to the travel route.