After interrupting "preparing backup" several times, unable to see time machine in finder

Sorry, you have lost me.

Recently it has been unable to backup; it's stuck in "preparing backup".

You never want to stop "Preparing Backup" assuming there is plenty of space on the Time Capsule for new backups. If you do, Time Machine has to start all over from the beginning of the process of checking all the files to be backed up....before.....it will start the backup process.

Let "Preparing" run until the backup has completed, or until you see an error message appear. You might see "Preparing for up to 6-8 hours or more. Let it run. Come back the next day to check on things.

After restarting and talking to Apple, my Time Capsule is still showing up in Finder as "shared", but Time Machine no longer appears under "devices"

Time Machine is not a device. Time Machine is software on your Mac. If you are running macOS 10.13.x the Time Capsule.....(not Time Machine).....will show up under the "Devices" or "Locations" heading on the left side panel of any Finder window. I can't remember whether High Sierra displayed "Devices" or "Locations", but they are esentially the same things with different names.

If the Time Capsule does not appear under the "Devices" or "Locations" heading on the left panel of a Finder window, Time Machine will never a complete a backup.....because your Mac does not recognize the Time Capsule on the network.

This is all complicated by my using OS 10.13.6 (long story), and that the Time Capsule is full.

New information. If the Time Capsule is full or even close to full, new backups are not going to run.

BUT still nothing under Backups.backupdb.

Because the Time Capsule is close to full and new backups cannot run.

If it were me, and I wanted to continue to use the Time Capsule, I would erase the hard drive on the Time Capsule and try to start over with a new backup.

But....if the Time Capsule was 6-7 years old.....both the power supply and hard drive are entering the stage where either could fail at any time. I would retire the Time Capsule if it was 6+ years old.

Bottom line.....if the Time Capsule is full, or even close to full, nothing is going to happen in the way of new backups.

So, you'll need to decide what you want to try next.

and double-click on the sparsebundle

You don't want to be opening up the sparsebundle because it is very easy to corrupt the data in the backup doing this.

The Time Capsule is not visible in the Disk Utility app

It is not supposed to be. Disk Utility only displays drives that are connected directly your Mac, like a USB hard drive. The Time Capsule is a network drive. It connects to the network, not directly to your Mac.

Disk Utility does not display network drives, that is why you will never see the Time Capsule hard drive named "Data" in Disk Utility.

You can see the sparsebundle file in Disk Utility because you have opened the file and it is mounted on the Mac's desktop. Playing with the sparsebundle file this way is frankly an invitation to disaster though unless you know what you are doing.

Also, doesn't the Time Machine software delete old backups in order to accommodate new backups?

Yes, if the backup is a very small "incremental" backup. But, if the Time Capsule is full or close to full and you are trying to make a new backup of a Mac or add a significant amount of new data, Time Machine cannot delete enough data to make room for the new backup, so the backup fails. Apple is very quiet about this small detail of course.

the Time Capsule is full and old

If you don't want to erase the hard drive on the Time Capsule, just set it aside in case you need to go back and view or pick up a backup in the future. Very few users ever need to do this though.

You also have the option of connecting a USB hard drive to the USB port on the back of the Time Capsule and using the Archive function in AirPort Utility to copy all of the data on the Time Capsule hard drive over to the USB drive.

This would be the thing to do if you want to keep a copy of everything that was on the Time Capsule drive....before....you erase the Time Capsule hard drive to start all over with new backups to the Time Capsule.