The two blocks of time you have set up to deal with email are entirely reasonable. If you only checked email during that time slot, would you still feel anxious?

TBH, it sounds like you have good reason to feel anxious about the payment if they didn't pay the last guy, haven't paid you anything, and don't have any money unless you bring it in.

I find the whole not paid to be on call/expectation to answer emails stuff challenging. I'm contracted for a certain number of on-site hours per week and have no obligation to answer emails outside that time, but I can't stop myself from checking... and feel like there is an expectation that I will answer immediately. I don't have a panic response to email so much as some of the messages are infuriating and I don't want to feel lousy on my days off. My solution was to check work emails less often - no autodownload. I also implemented a minimum charge, so if I check my email, I bill them (which helps me not check so often).