This isn’t strictly software related, but a lot of us have to deal with the horror that is email. Email is not a good solution to any problem, but it is ubiquitous so it is used for all things: personal correspondence, commit tracking, defect notification, task notification, etc… Email is all to often used as a mechanism to pass-the-buck.
The only way to deal with this email overload is to set boundaries on email usage. A lot of people will find these boundaries annoying, if not unworkable. Just stick with it and lead by example. Eventually, in a time of high stress, you will be able to get to important messages fast and they will be left floundering.