Hey, Jon,
Getting rid of the "weekend woodworker" blade that comes with most saws can be a starting place, but if you've done that and the blade is still stopping under load, I'd wonder if the drive belt is tensioned correctly, or if the lock bar on one of the pulley wheels is gone.
Don't know the Sears set up, but the first thing I'd check because of what you're saying happens under load are the pulley wheel lock bars and set screws. You should have a pulley wheel on both the motor shaft and the shaft turning your bottom drive wheel. Both shafts have a slot that aligns with a similar slot on the inside diameter of the pulley wheel that slides onto them. Your drive belt rides in these pulleys. A small metal bar slides into the space formed by these two slots and is held in place with a small set screw. The bar & screw lock the pulleys against the shaft to keeps them from slipping. It is common for the vibration of the machine to cause that set screw to loosen, and oftentimes when this happens, it will "throw" the little stop bar causing the pulley wheel to slide on the shaft. When this happens, the pulley may maintain enough friction on the shaft to turn the blade, but as soon as a load is added, the blade will stop because the pulley is slipping and spinning loose around the drive shaft. Check both pulley assemblies to see if they are locked in place and the stop bar is indeed present ( it's an easy little piece to lose).
Let us know what you find.