Mom,
I have a similar TM as you, both in model and age. I have a 3500 lb axle, so I bet you have the same. I have had my TM for 9 years, and have traveled 10's of thousands of miles, including several transcontinental trips, those on some oven-hot desert roads, and some in rather evil, bumpy, rumble strip like mountain roads for long stretches.
Loaded, our TM weighs about 4,000 lbs, which is similar to what others with 2720s have reported. I've had several tire blowouts, but never any sign of an axle failure.
The load range C tire is what the factory put on. Woefully inadequate, in my opinion. I had 1 blowout with a C, granted on a hot summer day at the end of one of those transcontinental trips, about 20 miles from home.
Then I got the Kumho 857 tires, also a 14" tire, but a load range D, and thus a higher weight capacity than the OEM C tires. Had 3 years of good service out of them, but had a blowout.....make that 2 blowouts, on a Sunday afternoon in a relatively small town. Made the decision then and there to move up to a 15" wheel and a load range E tire. This required no real modification to the TM, since I already had the 2" factory lift kit installed (a $40 part); I did have to put on the washers behind the wheel skirt to move it out from the trailer a bit for more tire clearance.
I had the E tires for about 5 years, running at 65 psi after I learned that pretty much anything above that beats the trailer to death. I had them balanced, which I think is key, as others here also report. After 5 years, I replaced them with the same size and load tire, but different brand (Maxxis tire, see my signature). And so I expect another 5 years of faithful service.
So simply based on my experience and the fact we basically have the same trailer, I'd get a lift kit, install it if you're handy, otherwise get a shop to do it. It's relatively inexpensive ($100 labor maybe). Then get 15" rims (probably no more than $50 each, maybe less), and 15" tires. You could keep a 14" rim for the spare, and swap it out the next time you replace the 15" tires if you need to save money. I'm often out in the boonies with no help/tire shop in sight, so I got a 15" spare at the same time.
Dave