Your trailer originally came from Keystone with ST225 75D15 LRD tires. They were bias ply, probably rated at 2540 pounds @65PSI.
Almost all trailers now come with radial tires and I'd recommend you seriously look at upgrading tires to a 225 75R15 LRE (10 ply) which is the largest tire in that size range that will "comfortably fit in your wheelwells". They are rated at 2830 pounds @80PSI. While that may not seem like a substantial increase in load capacity, when you consider that your trailer's GVWR is 10375 and there's probably 1000 pounds being carried by the tongue, that leaves 9375 for the axles to carry. At 2830x4, you'll have 11320, about 2000 pounds of "excess capacity"... That's around the 18-20% recommended surplus capacity.
I would recommend Carlisle Radial Trail HD tires. They are among the best rated ST tires available, along with Goodyear Endurance and a "falling brand", Maxxis ST8008.
Endurance will run you around $140-160 per tire, Carlise is available at WalMart for $94 each
https://www.walmart.com/ip/seort/55012166 Whatever tire you finally choose, have them mounted AND BALANCED !!! Many people fail to recognize the importance of properly balanced tires on a trailer where there are no shocks to help absorb/dampen the road hop and bounce created by unbalanced tires.
In the past 5 or 6 years, I don't recall anyone having problems with Carlisle RT HD tires.
You've got to realize (maybe you already do) that your trailer was built during a time when tire safety/safety margins were not a priority. Your trailer came with tires that barely could carry the axle load, much less survive the age degredation they encountered. I'm surprised that you made it as long as you did on those tires and would strongly urge you to replace them and possibly the wheels (depending on condition and their weight rating) before towing your trailer again.