You need to deauthorize the old ID first, then authorize with your new one.
Deauthorization is a hidden feature: ctrl-shift-D to ADE (cmd-shift-D if it is a Mac).
A well implemented program would warn about deauthorization of an old ID when you tried to authorize with a new one, and implicitly remove the old authorization if you confirmed.
A poorly implemented program would have a clear deauthorize option; probably in the menu next to the authorize option.
An attrocious program would have a hidden option that you need to resort to external help to find; ADE is sadly an attrocious program that Adobe does not appear to have any interest in improving.