> I then exit my current FF, and switch to it, and back again.
FWIW, you can also run multiple profiles simultaneously. They are independent processes, sharing no resources or permissions.
This is my model for difficult sites. If I'm really concerned, I use FF network config to allow access only to the domains I think are proper.
Although in the case of banking, I prefer to use the official mobile apps. Some are actually pretty good. Others are awful. But I trust the iOS app sandbox and I trust my banks.
I also block traffic at the network level, so if the bank app attempted something egregious (e.g. tracking via the basket of Internet deplorables), it would fail.
My fix was to create an entirely new profile, with no customization, no cookies restrictions, no add-ons, and use it only for financial sites.
I then exit my current FF, and switch to it, and back again.
All my issues vanished after doing that.
You could also create a different user in Linux, and isolate that way.
Hope it helps.