I have a American Standard toilet that I think is around 17-20 years
old. It has the ball cock/flush ball mechanism in it.

Recently, whenever the toilet flushes, after the water refills the
tank, the flush ball arm vibrates very slightly, but it, makes a huge
racket.

Is there a way to quite that vibration???

On the same subject, our downstairs toilet, (brand new) has a
different kind of mechanism. There's no ball cock or flush ball. It
looks like a completely enclosed valve or something.

I'm wondering if it's possible to install a mechanism like that in the
upstairs toilet.

Anyone have any thoughts??