Identifying and writing about it is a good first step. Some people need to talk these things through (with a therapist), or talk and practice behaviors (CBT), some people can write it out. For me, writing out the anticipated experience makes material changes in my processing. It works kind of like this:

Follow the thread you started here and journal about the anticipatory experience. Use your imagination and write about it until you don't have any more words. Get yourself to feel the fear more directly than just anxiety, and lean into the fear, feel it, let it inhabit you. Once you are feeling the fear, actually opening an email will not be a challenge, because you are already in a place where it feels like the worst has happened. Maybe the email(s) match your anticipation, most likely not, but nevertheless there likely is something to process from the email, to work through your reaction as to whether there is an obligation. Continue to journal about this. Take all day if you have to.

The process of getting it out as words serves a bunch of purposes. It helps your meta/reflective cognition process the anticipation/anxiety as an explicit mechanism over which you have a choice, and build an understanding of whether it is a worthwhile expenditure of energy. Almost certainly you will arrive at the understanding- not an intellectual understanding, but an experiential one, because you experienced the fear, and the reality, and you reflected on it- arrive at the experiential understanding that this is not a good use of energy, and at that you will feel tremendous relief. Next morning when you are presented with the option of opening emails, you will recall this experience that the anxiety juice was not worth the squeeze, and it will be easier.