The "plastic piece" is known as cartridge and must be replaced to eliminate worn wobble. If you can identify brand and model,new cartridges are available at plumbing supply stores and a few hardwares. But,yes the water must be off while changing cartridge.

Thank you. I actually thought that was the name based on some useful googling. Thank you for giving me a word to sound like I know what I'm talking about a little.
Is there any trick to identifying the brand and model or is that pretty much impossible?

Not easy without experience. Most diyers take the cartridge with them shopping. Have a plan B in case you get in trouble. Don't take it apart late Friday or on weekend. It doesn't require a licensed plumber, a good handyman will do fine and cost less.

Sounds like a plan. Any not so good temporary way to minimize that jiggle?

Considering your lack of experience,I wouldn't reccomend you try short cuts unless you already know where to buy a replacement. (:

Good advice. Just had to ask. Thank you for the help.