Setting cut depth on a circular saw

If you're sure it's the shaft that's loose, and not a misalignment in the blade attachment, then it's most likely one of the bearings supporting the shaft that's bad. Check for play in the bearing flange (Component 36). If that's solid, then check the bearing sleeve (component 13). Unfortunately if it's the armature that's damaged (component 3), then you're better off buying a new saw.

Back in the day when I used to put up vinyl siding, I found that the best way to cut it with a circular saw was with a cheap 140 tooth plywood blade installed backwards (to give the teeth a negative rake and prevent chipping):

For really hard plastics, I've used a diamond abrasive blade before and that did a great job (these are incredibly expensive if we're talking about a single cut though). I would imagine an abrasive blade for cutting metal would do just as well though:

You should be able to pick up either of these for less than $5.

Wear good safety glasses and do a trial cut first with whatever option you go with - plastic will chip long before it will bind the blade in most cases, and these can be both sharp and high velocity.