I've been working with my host to figure out some outgoing email issues... I have close to 1000 items in my current Background Process. Most of them seem to be "Sending a batch of PWA notifications".

Is there a way to stop/clear this queue from processing and sending more notifications until my email issues have been fixed?

There is no way in which to temporarily pause background tasks unfortunately. Removing them entirely may be an option if its needed, but we would usually recommend resolving the initial issue and letting them run through, as you are doing at present with your host