Once my account is shut down, how do I preserve the legal value of its email?
Shutting down does not necessarily imply deletion of messages. It is unclear what that means in your specific situation. But one suggestion is to create a backup (in local storage and/or forwarding the message) and also make the institution a specific request for preservation of evidence.
Your request should be made by email or having the institution sign a receipt copy. Otherwise you are giving the institution the opportunity to falsely deny that you asked that records be preserved.
I'm afraid that backing them up in local storage won't do, as in litigation it could be claimed a forgery.
The institution would have to prove its allegation of forgery, which will be more difficult to accomplish if it received the aforementioned request. During discovery --or its procedural equivalent-- you can require the institution to produce the records you asked it to be preserved. That will make it easier to corroborate authenticity of records unless the institution itself indulges in forgery.