The Sears unit that you have was made by Generac, which is now owned by Briggs & Stratton. There's a possiblity that it may be a minor repair, but that would need to be determined through effective troubleshooting by a qualified generator technician.
When you say it's no longer producing, are you certain about that or are you making that determination because there is no electricity at the receptacles? It's possible there may be a bad outlet, breaker, or wiring that would cause this. A relatively inexpensive repair.
If there's no current being produced at all, it could be because the rotor has lost residual magnetism that can be restored by "flashing the field", or it could have a damaged or worn set of brushes that can be easily replaced. Or even the bridge rectifier board (about $60) could have gone bad. If it's any of these items, repair would certainly make cost-effective sense, since you say the engine is still running well.
However, if the rotor or stator has gone bad, the rotor is obsolete and the stator is over $500. It would not be work repairing if those components are faulty.
I would suggest this simple first step: with the generator off, remove the slotted squarish cover at the back of the generator that's held on by four screws. You will find a couple of wires going to a set of brushes in a plastic housing. I would check the brushes to see what they look like and also inspect the slip rings on the rotor they run against to see if they are dirty. They can be carefully cleaned with fine-grit sandpaper if they are dirty. If you remove the wires from the brushes, pay attention to which wire goes where -- one is positive, the other negative. They have numbers on them to make it easier.
I hope this info is helpful!
If you're close to Mesa, I suggest Grover's. Talk to John Grover if you can - he's one of the co-owners. I've known him for years, and have a great deal of respect for him.
Grover's Lawn & Garden Power Equipment Company 130 W Hampton Ave Suite 9Mesa, AZ 85210
Very best of luck with it, Mr. Hudson.