I had the same problem with my Model B. When I took the tire off the rim, I found that the salt solution in the tire had leaked and corroded the inside of the rim so badly that it had eaten small holes through the rim and had badly corroded around the valve stem hole.
I had the rim sandblasted and repaired the holes in the rim with J-B Weld. I put a piece of tape on the outside of the rim and filled the holes with J-B Weld on the inside. I then sanded the filler until smooth and primed and painted the inside of the rims with two coats of gloss enamel so that there would be nothing to snag the new tube.
I did not have the salt solution put back in the tires because I will not be using the tractor to plow, etc., only to run a rotary and finish mower.
The rims look new, and I have had no problem with punctured tubes or the J-B Weld coming off the rims. It appears to be a permanent solution in my case.