Emails can be traced down to the computer that sent them. Yes it also carries the IP address. From an in-depth search you can figure out which routers it went through to get to its location.
In short yes it can be back traced to a sender. Even if from a public library, there are probably surveillance cameras.
Sorry, but the email sounds Fake! At least one soy boy would know tracing isn’t that hard. They are stupid but just not that stupid.
Real question is - Why did Ashville Police send this email to themselves?
And don’t you need a library card to use the computers?