Seagate backup says I can't save changes to files on the disk

You can try running a Disk Utility Repair/FirstAid on that drive when it is not busy. It may have a minor problem that can be repaired.

But I suggest you may want to ADD a drive, rather than replace that drive.

You can ADD another drive to Time Machine, and it will begin a new stand-alone backup set onto it, starting with today. Then every-other backup will be done using every-other drive until one is removed.

Without another backup set, the Genius Bar can't do anything magical. they will just tell you its not working, and you already know that.

You can contact Apple support for a Genius Bar appointment, but that is usually for a problem with an Apple device - not a third party drive. Your problem may be with either, so do contact them. Having said that, drives can fail at any time, so you may have to buy another.

Download and run DriveDx, post the report for the Seagate drive here.

From that, we see what that tool reports for the Seagate drive.

Hard disk failures can and do happen, as can volume corruptions.

If you’re particularly concerned about the availability of your data, running two Time Machine backups in parallel is fairly common choice. With that, you can have some depth of backups maintained, even if one of the backup drive fails. Or even if Time Machine gets tangled. Which can happen.