While you wait for a knowledgeable reply, I’d suggest asking Verizon again.
I’m a long time Verizon customer and generally satisfied. But, on several occasions I’ve gotten extremely different answers to tech questions when I’ve asked different reps.
Sorry to be cliché, but there’s (probably) an app for that. I used one to download a text message conversation from my Android phone to my computer. I’m sure you could probably find one to do what you need.
Best I can do is “maybe”. Motorola phones, for instance, may allow you to download the messages via Motorola Phone Tools (MPT) I’m not sure what format they end up in.
If you can find out the make/model of the phone, I might be able to ask around.