My Seagate Backup Plus external drive is apparently full and will not delete earliest files so that it can continue. What should I do?

I presume this is a Time Machine drive. If you aren't using iCloud for storing your Photos and documents I advise you do so.

You have two options:

1. Format the drive and start again, this will sacrifice all your backups.
2. Remove the old drive, store it and purchase a new external drive ( at least twice the size of the boot drive)

Time machine doesn't have to be a sophisticated drive and it is best not to install any unnecessary drive software which came with the drive.

I am using an old 1TB hard drive (mechanical) from my previous Mac and even though it is only USB 2 it works quietly in the background.

For what it's worth, this seems to be a fairly common complaint on this forum. To date I have not seen an actual solution to the problem, except to erase and begin again. this can certainly address the problem. My solution is to use another, better backup app such as Carbon Copy Cloner which seems to function without any problems. It also has more features for restoration. It also backs up all email files, which Time Machine does not do for some reason.

I personally do two backups. I use Carbon Copy Cloner to do a backup one daily and I also do another separate backup manually. Each time I add or change a file I copy the new file over to the second backup disk. This way I am backed up to the minute. Because they are not that expensive to purchase, I have two backup disks.