Transporting 31' Class C

That's exactly what I am worried about, the height restrictions.

Look for Low Boy service. It will probably cost around $95.00 to $100 per hour
cost calculated from their home base to your place and then the haul and back to their home base. They might need permits for height depending on how tall it is loaded on the low boy, permits are extra cost.

I am trading it in, and want to avoid putting 8 tires on it if I can, or at least 6 if I don't do the fronts if it will get towed.

I called the company that moves my construction equipment, but they do not travel far from their shop. I'll try a large towing company I know in my area and see if they can do it. It is about 70 miles one way, so I know it will not be cheap, but it should be cheaper than trying to make it road worthy, as it has been sitting for quite a few years.

Just web search "heavy wrecker" and you should be able to find outfits with a Landoll trailer

Just a suggestion but tire stores are everywhere. It would probably be a lot cheaper just to buy tires near where the RV is. Another alternative would be to remove wheels and take them to a tire store to replace the tires. It is hard (expensive) to find a flatbed to haul something that big.