Turns out, the throttle plate shaft was broken and the plate eventually just got stuck. That would account for the governor not being able to regulate the engine under load, and also explain the stuck throttle. I'd replaced the throttle plate recently after it had lost a screw and wanted to make sure I didn't lose another one, so I must've overtightened the screws and weakened the shaft.
So I replaced the carb yesterday with a Stens unit from Amazon, and all seems well, though I'm seeing a bit of surging/governor hunting for the first minute or two. Hopefully that'll work itself out since the Stens is a fixed-jet carb with no accessible mixture adjustment screw (not that I'd know what do with one, but I do have the engine service manual that explains the procedure).
>> Have you ever pulled the metal engine shroud and cleaned the top of the block and cooling fins? This is a yearly maintenance item in ALL engine manuals. Neglect this and you are looking at engine damage.
Your sig led me to pull the engine shroud yesterday, and I found the crankcase covers' frontmost four screws had backed way out and leaked oil all over the top of the engine (I don't think I was too far from having a couple of those bolts back out far enough to contact the flywheel). I cleaned everything up, added some Loctite Blue to the bolts, and will definitely make it a point to pull that shroud every spring during maintenance. Thanks for the tip!
I got the back cover open on my tach. Simple coin cell battery inside. Sucker is 10 plus years old and still has a visible readout. She is dim but if you angle it right, can read it.