Hi,
From our records, buttpillow.com and old.buttpillow.com are separate sites, correct?
If so, you will need to access old.buttpillow.com’s WP-Admin, and disable UpdraftPlus (or set the schedule to manual).
Then, ensure that the schedule for backups is set up on the new site.
If you still find that the old site is being backed up instead of the new site, please could you send us a copy of the latest backup log?
The contents will be too long to post directly, but you can use an online service such as Dropbox or Pastebin, and post the link here.
Best Wishes,
David N