Since you have already tried standard troubleshooting, this issue is likely caused by a communication conflict between Apple’s Live Voicemail feature and Verizon's servers.
Try this
Turn Off Live Voicemail Temporary/Permanently
Apple's Live Voicemail handles calls on your device before they ever hit Verizon's system. This frequently causes custom greetings to fail to save or revert back to the default message.
- Go to Settings > Apps > Phone (or just Settings > Phone depending on your exact version).
- Tap on Live Voicemail.
- Toggle it OFF.
- Go back to your Phone app, tap Voicemail > Greeting, and try recording and saving your Customgreeting again.