Did you go to Phone/Voicemail, tap on Deleted Messages and cleared them, while the phone is connected to Cellular Data?
The "full" message comes from your carrier's voicemail server, it does not mean that the phone is full, only that you have reached the limit on your carrier's voicemail server. Normally when you delete a message on your phone it goes to the Deleted Messages folder. This does not delete them from the carrier's voicemail server until you clear them from the Deleted Messages folder. And the carrier's voicemail server is not cleared unless you have cellular data enabled.
You can also call into your carrier's voicemail (check their instructions, but it is usually done by calling your own number from your phone and following the prompts) and delete the messages directly there.