Shower faucet

First trip out, first shower.  Cold water handle broke off.  Can’t see how to remove.  CHEAP plastic.  Any experience on how to remove, replace?

There should be a cap in the center of the handle, right above where the valve shaft is.  A pocket knife should be able to pop it off (don't let it fall into drain!).  Under that cap is a screw.  Remove screw and handle should pull off.

Not  that kind of handle.  This cap is shaped like a dome and then makes a 90* bend out to the handle.  Dome and handle are(were) one piece.  The dome is loose over the valve, but won’t slip off, like the bend is holding it from coming off?

Can you post a picture?  That doesn't sound like the original shower handle.

That type of handle typically has a small hex head set screw accessed from below so you can't see it.
Or not...

The sink and shower handles ends are of the same shape so they probably original.    The sink handles does have “caps” that look like they could be pried off right in the 90* bend, but the shower does not have them.  The hot and cold handles on the shower have a very thin red/blue washer.  Where the washer is, is where it broke.  While typing this I played with the hot side and found the handle will unscrew from the “domed” part.  There is a splined “tube” that is threaded on the inside that the handle screws onto.  The spline part is where it broke.  Will try to get some needle nose in the hollow and unscrew the broken part out.  Right now I can still operate the cold side by rotating the dome.  There is a faint ID on the top of the housing, the number 60 or 09, a symbol that might be SA and something that looks like a old time shield with a slash thru it.

The sink and shower handles ends are of the same shape so they probably original.    The sink handles does have “caps” that look like they could be pried off right in the 90* bend, but the shower does not have them.  The hot and cold handles on the shower have a very thin red/blue washer.  Where the washer is, is where it broke.  While typing this I played with the hot side and found the handle will unscrew from the “domed” part.  There is a splined “tube” that is threaded on the inside that the handle screws onto.  The spline part is where it broke.  Will try to get some needle nose in the hollow and unscrew the broken part out.  Right now I can still operate the cold side by rotating the dome.  There is a faint ID on the top of the housing, the number 60 or 09, a symbol that might be SA and something that looks like a old time shield with a slash thru it.

If your faucets are like some I have at home, pictured below, the lower part of the "dome" unscrew to remove, also pictured below with my finger pointing to the section  you twist to unscrew.