Hi melnick03,
Welcome to the Apple Community.
Thank you for posting your info. here in the Apple Community.
Seems it is best to just have one Apple ID account for everything to do with Apple, and to keep it up-to-date.
Re: "An Apple ID I haven’t used in 15 years is coming up"
" I have no idea what that password is"
Wondering if the old Apple ID has still been in use somewhere through the years ... eg: Messages, iTunes, or FaceTime ... etc.
If so, you could likely retrieve that password from the app.
If not, and you want to retrieve the old Apple ID password:
What to do: If you forgot your Apple ID password – Apple Support
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Re: "cannot retrieve it since I no longer have access to the email".
Wondering if you remember the old eMail address so you can use it to log-in to the old account, if need be.
If you forgot your Apple ID email address or phone number - Apple Support
or
Enter information to find your Apple ID:
https://iforgot.apple.com/appleid
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Else:
If you should want to delete the older Apple ID there may be things to consider:
How to delete your Apple ID account - Apple Support
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Congratulations on the new Apple Watch!
Re: " ... try to set up my new Apple Watch"
"my “current” Apple ID is still listed in settings".
Apple Support's article:
Check your Apple ID device list to see where you're signed in - Apple Support
eg : on scrolling down : says how to: "Add a new device to the list"
If needed:
Apple ID - Official Apple Support
All the best :-)