It sounds like the tires are worn. They should be tightly on the wheels, and the blade should ride with the teeth sticking out over the edge, not centered (unless it's a 1/8" or maybe 1/4" blade.)
That saw's worth keeping so I'd start out with replacement tires and new blades, try Timberwolf low tension 3/8" (or wider) 4 tpi skiptooth for resawing.
You should also tune the saw. This entails adjusting the blade guides such that they are almost touching the blade, adjusting the thrust bearing such that it's not touching the blade until the wood makes contact, adjusting the blade tracking such that the blade's teeth aren't riding on the tire but rather hang off the front a little, and clean the wheels and tires (remove built up sawdust, pitch, etc.) When you're done cleaning you can stick those tires down with an adhesive if you're going to use the old ones.
My little Delta's top tire kept working its way off because the blade teeth were too far off the front and the tension was too high. It simply sqeezed that tire like toothpaste out of the tube.
Good luck.