great river road

Best to use interstate unless the route is free of snow.

That depends, if you want to get to your destination in a hurry, or if you want to see some great sights along the way. All you need to do is watch the weather and drive accordingly. Even after a heavy snow, the River Road will be cleared the next day.
Obviously if you watch the news, you know that they also close the interstate highways when things get too bad. We have driven the River Road in the Winter and it was no problem at all. Be sure to call to check on your anticipated stops to make sure that the RV park is open and that you can get in to a parking place if there is any snow. Some RV parks do not clear the snow very quickly and others do.

Just follow the river! If it snows there will be some on a highway or back road! at least until central IL.

We live 10 miles from the Great River Road in Illinois. We wouldn't take this route in the winter time. You can't really see much of the river and it would be mostly all two lane roads that may or may not be cleared good after a heavy snow. Also there would probably not be many campgrounds open in the winter. The best part of the road that we know of is between Nauvoo and Hamilton, Illinois. We do take this when we can.

Lot's of nice scenery along the Great River Road, but slower going and no RV parks available until you get south. If you are staying in the south until spring, take the GRR back north instead.

If you run the west side of the Mississippi watch your speed in Lake Providence LA. According to the nice man in a uniform carrying a gun, a 40DP towing a 3400 pound truck can get to 45mph in a 25mph zone from a dead stop in less than one city block. Needless to say I had to pay the local out of state transit fee to the local court of $181 in November 2008. "Time" named this small town the poorest place in America in 1997. If you google "Louisiana speed traps" Lake Providence in right up there. We haven't driven that section of scenic US61 since then.

OP:
Use www.rvparkreviews.com to find and see rating of RV parks of all kinds in your travels South.