Frustrated and need help!

[joesheets]

I am so frustrated! I am including a pic of a part that has failed in 3 of my faucets and I cant seem to find out where to get a replacement. It is some kind of valve that is in each handle. The problem is they lock up and stop working.
Any help would be appreciated!
Joe

 

[ExtraMilePlumbing]

I don't know wny it's locking up but I do know that after the third time, I would just replace the whole valve , sorry if I ain't much help

 

[joesheets]

3 different valves have seized up on 3 different faucets. Once they seize, they don't work so I am looking for a replacement, I just don't know where to find them to replace them. That was the point of my post.
J

 

[Caduceus]

Good pictures of the cartridge, but let's also get a look at a few pics of the faucets in question that show the handles and spout.
I should add that the spline count might be needed. Count the number of little ridges on top of the cartridge stem by looking straight down onto it.

 

[joesheets]

Here are the pics of the bar sink faucet and the bath faucet.
There are 16 splines.
J