There is no automated help from Adobe or others on this.
- Find the .epub file(s) for the book on the old computer. (or if you are unlucky, the .pdf files)
It may well be in your Documents under "My Digital Editions"
e.g. C:\Users\<userid>\Documents\My Digital Editions - Copy the files using whatever method is convenient (USB stick, network sharing, ...)
Probably easiest to copy them to the equivalent folder on the new machine. - Open ADE on the new machine: one of
ctrl-O or File/Add to Library in ADE
drag-drop from Explorer onto ADE
double-click in Explorer (if ADE has registered with the .epub filetype)
As long as you have used the same AdobeID to authorize both computers, all should be fine.
It's the underlying ID that matters, and changing the password does not change that.
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Also note that there is no automated help for backup either.
Neither Adobe nor the publishers guarantees a copy of you books somewhere in there servers.
Make sure you have a safe backup of all your DRM .epub files in case of computer failure.