Yet another Toto Drake sound

Lately, our 4+ year old Toto Drake has been making a strange rhythmic sh-sh-sh-sh sound, reminiscent of a rotating sprinkler. This occurs many minutes after a flush has occurred and (seemingly) finished. Usually it's a pretty quiet sound, but sometimes it switches to a very loud version of the same sound, which lasts for a few seconds and then all noises cease. The noise definitely appears to be coming from the fill valve.

Any ideas?

If it matters, for quite some time the toilet has seemed to have a leak somewhere: If you pour a cup of water into the full bowl, it sounds like a cup of water is slowly leaking out. Apparently into the drain pipe, since there is no sign it's going onto the floor or otherwise outside the pipes.

The fill valve needs a new $2.47 assembly cap

The part can be replace without tools in less than a minute.
The Korky Quietfill valve is quieter than a Fluidmaster and doesn't have water hammer like a Fluidmaster.
So, if I were you, I would just pop in the $2.50 part.

As far as the bowl goes, if you add water to a full bowl, any excess will drain out the trapway.
If you look at the top bend, that's as high as the water will ever get.
The water goes up over the dam, and down the drain.

OR you can replace the fill valve with whatever you prefer to use in your area for a fill valve... (I prefer the fluidmaster 400a) ???

dances with pumps