Verizon, you're seriously the worst sometimes.
whatever123ABC, I know it's been years since you posted your question. It's probably the only reason you created your account. I too had this question and found the Verizon response lazy and less then useless (no one wants verizon cloud). I made this account just to post my response in hopes it helps the next poor soul who runs up against the same issue.
Saving messages in message+ is not hard it's just not obvious. By default SMS messages get backed up to the VZMessages folder. Instead, I'd recommend saving it to the Downloads folder or somewhere else. Tap the conversation to select it (or multiple), and hit the download button. From the download screen you can click the caret icon in the top right (optional step 1, shown below) to navigate up the directory hierarchy and select something like Downloads or Documents (step 2). Finally give it a name and click save (step 3). From there the Files app or one of the common file storage apps can pick it up and save it or email it or whatever you like.
Obviously if there's an update in the future these steps may be slightly different. Good look y'all.
