Toilet update for those interested - went to Lowe's yesterday (and used sanitizer when I left!) and got a new fill valve. Now I can't get the line from the wall to sit properly on the new valve, because it is metal instead of flexible, and it apparently came with the house, since it's fused into the wall good. Gonna try to WD40 it and see if that will loosen it so I can replace it, but if not, I'll have to put in a call to my plumber friend.
Before you start wrenching on that valve I recommend you turn off the water to the house in case you break it off. Just sayin'.
The water to that pipe has already been shut off, naturally. But I'm not going to go full bore on it. Not even sure I could, because there's precious little space between the toilet and the corner.