how to tell if oldest backups are deleted when backup disk is full

FYI ... I just double-checked following your steps to delete local snapshots on my 2010 MacBook Pro, running macOS High Sierra (which what appears the OP is running on their Mac) and those snapshots do not appear in the Disk Utility at all. I verified using tmutil listlocalsnapshots that they did exit on my notebook, but not in the utility. Am I missing something?

Looks like for earlier versions of macOS, you only have two options to removing these snapshots:

1. Remove them by disabling TM, wait 10-15 mins., then re-enabling TM.
2. Use the tmutil command to delete them one-by-one.

How much free space does the Finder say os pm the TM drive?

If there's not enough room on the boot drive to create a local snapshot you'll get the messages tha there's not enough room on the Mac drive.

You can use Disk Utility go into the boot drive and delete load snapshots.

Experience has shown that Time Machine doesn’t delete the oldest backups. You have two options, either reformat the drive and start again or a better option, now that storage is cheap, is to unplug the old disk as a fall back and attach a new bigger drive.