So, like you'd expect, I decided to take them apart and fix them.
Taking the stove apart is nothing new: been there, done that. The individual burner valves were a little trickier. They're bolted to a manifold with gaskets on either side. Taking them out was actually pretty simple, it was basically one bolt with a 1/4" hex head.
This is one of the valves as removed. Pretty much gunked up with grease and crap.
And the same valve disassembled. The whole thing is held together by the two small bolts.
Some rust-penetrating lubricant works wonders to get the grease and gunk off, particularly with the judicious application of some tapered cotton swabs and toothpicks.
The cone-shaped piece is the main valve mechanism; it fits inside the larger body and the spring holds it in place with just the right amount of pressure. The problem was primarily that the cone-shaped piece was binding.
When the valves are new, they are coated with a thin layer of a high-purity silicone grease. It's the same stuff that's inside a fluid-head tripod, and oddly, it's also the same stuff used in some water valves-- which meant I was able to find it at the hardware store. Very little is actually needed, just a thin coat, about the equivalent of half a drop of water. Another good use for the tapered swabs.
The only trick on the reassembly was adding some high-temp silicone to the gaskets to help with the seal. It's the same material that the gaskets are made of, so it fixes any little rips or tears that might show up.
A little over two hours of work, and the stove is nearly good as new. The valves feel awesome, the ignitors work as expected (more or less-- they've always been a little crappy), and there are no gas leaks. Should be good to go for another few years.
Total cost was about 70 bucks for the new switches and electrodes, and a buck fifty for the silicone grease. I'm okay with that.
Also managed to make dinner, do another load of dishes and a full cycle of laundry, and even folded clothes.
I can sleep well tonight.