Sometimes ADE gets its registration/activation confused and in a semi-authorized state.
Unfortunately, it often then gives misleading error messages about what is wrong.
A common incorrect message informs you that the ID is already in use on another computer and cannot be reused.
This can often be resolved by completely removing any authorization using ctrl-shift-D to ADE (cmd-shift-D if on Mac).
Restart ADE, and then reauthorize with your (old) Adobe ID.
Sorry, this is probably a silly question, but do I have to be on a certain screen in the program when I press ctrl + shift + D? Also, do I do this on my new computer? As I do not have access to the old one anymore.
Sorry, I should have been more specific, you need to be on the Library screen of ADE for ctrl-shift-D.
That will give you a prompt to confirm you do want to delete your old authorization (you do).
This is on your new computer; it is that one that isn't letting you authorize for some reason.
It would actually have been good to have done it on the old one as well.
You will eventually run out of authorizations (6 devices), and then you will need to get Adobe to reset things.
Deauthorizing the old computer would have put off that day; but failing to deauthorize won't cause any other issues.
Often people can't as the old computer is either dead or given away.