Bless you MAGICGREY. Your solution was the only one to fix the issue - thanks!
Does not work. Deleted messages later reappear.
Mine has been working for quite a while now. Someone at Verizon finally explained that the problem is related to the Verizon voicemail service not interfacing properly with the Apple voicemail icon you see on your phone. Try logging directly into your voicemail on your phone--not using the voicemail icon on the phone. Delete your messages directly from your voicemail. See if that fixes it. You might have to do it more than once. There could be some update from Apple or your carrier that throws things off again for a while, but mine has been working well for months now. It seems there are a variety of causes of this problem, and a variety of fixes, some of which work for some people or some phone carriers, but not others.
IOS 13.3.1 iPhone 6s
I tried the previous examples and none worked. I went to the Settings/Phone and turned off Wi-Fi Calling and Calls on Other Devices. Then went into and deleted the the unwanted messages. It also let me go in and permanently delete them when it previously wouldn't. I turned both back on and all is good!
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