You don't indicate what size the TM backup (target) drive is, nor the size of the drive you are backing up. Normally the TM backup drive should be ~ 4x the size of the drive it is backing up. If you have the old computer backups plus the new computer ones all on the same drive ... then that drive could be getting quite full from that combination.
Normally older backup files are deleted but the backups are stored as incremental backups, just changed files are copied. This sometimes means that enough files cannot be deleted because the older files that haven't changed in a while need to be retained in order to have a complete restore for a later backup. This can be greatly exacerbated by keeping the old computer backups there on that backup drive.
Best plan here is to keep the old backup drive as an "archive" backup and obtain a new drive to start a new backup. The new drive should be ~ 4x the size of the drive you are backing up. If cost is an issue, consider using external mechanical drives for backup drives, they are relatively inexpensive and after the first long backup completes, the incremental ones should be very fast (mine take seconds to minutes).