After looking at some of the PDF documents for this model it does have exactly what you say "a carb".
CY: When I got mine that "runs fine" the carb clogged. I dissembled the carb, and it ran for another ten minutes. Finally I put in a fuel filter, which did the trick.
I don't think it's a carb problem, but could be. I'm inclined to believe it's a fuel filter problem. The PDF documents also show this. Most likely I didn't see this when I pulled the skins.
CY: Might have a filter. If not, I was able to cut out some fuel line, and put in a universal small engine filter.
As far as the exhust cleaning I was thinking baffles. I don't think it has baffles. And I don't think there is an air filter.
CY: Mine has a muffler of some kind, and no air filter.
A quick replacement of the fuel filter(56-6360) should tell me something. This is a Toro part number.
Do you think a generic part from HD or local hardware store could be used. Ordering just this part will cost more in shipping then the part.
CY: I'd measure the ID of the fuel line, and the diameter of the fuel filter you find at the hardware. It is very much likely to work fine.