I opened up the carb today. The bowl, at least, looks clean. I also looked at the fuel filter (none in the tank--just in-line) and it wasn't visibly gunked up, which I assume it would be if the oil had leaked up through the carb into the line.
On a related note, I went to talk to a neighbor who I know has a mower that is out-of-commision. Supposedly a starter issue--engine runs fine. I asked if I could take a poke at it and see if I could get it running again, and if so, whether he'd make me a good deal on it. It's a 50", 22 HP Toro, so it'd be a real step up from my current setup. I could hear the solenoid click, but the starter didn't spin. I got no voltage at the starter lug. I pulled the starter and hooked it up directly to the battery. At first, it didn't spin, but later it did. I turned it by hand a little, so maybe I loosened it up. It felt pretty rough while turning, but I don't know if it's supposed to feel that way. The other thing that was unexpected was that the worm gear was fully extended, and stayed that way. I'm not an expert on starters by any means, but that doesn't seem right to me. It seems odd that the starter would always be spinning when the motor was spinning. I also pulled the solenoid so I could get a better look at it, and sure enough, 12 volts on the battery side (as expected), a thunk when the ignition is turned, but no voltage on the starter side. Jumping the starter directly off the battery, I could get it to turn after I freed it up by hand, but it was really weak. At first, I thought the battery was low, but I tried several different batteries that I was sure were fresh, and none of them could get the motor to turn over decisively.
Currently, my diagnosis is, at least, bad solenoid, and probably also bad starter. I don't really know how motors work, electrically, but I have a hunch that a failing motor could cause excessive current draw, thereby burning out the solenoid. Is that right?