We spend a lot of time in Edgewater and have made the trip probably a dozen times over the last five years. If we were making the trip in January, we would check the weather for the next couple of days and look for night lows below 35 degrees.
If the temps looked good, we would run 66 to 81, then 81 south to 40. If temps were going to be too low or the weather was lousy, we would take 95 south to 85 south to 20. We avoid 10 in Louisiana at all costs.
Each morning we check the weather for the next couple of days and make adjustments. We drove it in December 2 years ago and ended up on 10 through Texas because of snow. I wouldn't be stubborn about which route during winter.
The drive can be a lot of fun or a tedious slog through cold, windy, dreary days.
We are in Arizona right now and the nights are chilly (30s). On a lake with better weather, check Lake Havasu.