Actually, I figured out the problem with Samsung's help. And this is one of the reasons why I dislike bloatware.
I have Verizon. The Galaxy S7 comes with their version of text messaging called "Message+." I didn't like it, so I put the default message app to the Samsung stock version.
And THAT was the problem. When I disabled Message+, the notifications stopped. When I made Message+ the default again, the notifications came back. It's aggravating to have to use Verizon's version but, for now, it fixed the problem.
So, if anyone else has Verizon and has the same issue, check what the default is. I'm hoping there's a way around it, but, for now, it will have to do.