New update causes website problems

If you haven't rebooted since the issue started, I'd do that first.  We don't make you exit Flash Player during an upgrade, but that can mean that the old copy of Flash that's in memory is still running, and things can occasionally get weird.  It's the cost of convenience.  Rebooting the machine should rule that out as a possibility.  I'd also recommend checking to make sure that the latest Firefox is installed.

That said, I'm not seeing any Flash content on Zillow or Trulia.  TMZ makes an absurd number of advertising-related requests (925 initially, and then pings back continuously as you look at the page), and some of that content is one-pixel tracking stuff that uses Flash Player.  The embedded videos and images are generally not Flash.

If you go to about:crashes in Firefox, you should see a list of crash reports.  If you click on one (or better, right-click and choose Open in New Tab) the name of the link will change to start with "bp-".  If you can submit a couple, and then just post the report ids (bp-xxxxxxxx), I can usually take a look crash dump and just tell you what's causing the hang or crash.

Shockwave Player is different to Flash Player, so I moved your question before answering.

There is a report and bug logged here:

Flash Player 22 - Flash content does not display since player 22 install

Can you read the description and see if it's similar to your issue? If it is you could add more information to the bug report here:

Bug#4165783 - Flash Player 22 - Flash content does not display since player 22 install