I am also having this problem on my S7. I will delete about 20 conversations a day, which is already a time consuming process..I need my phone for texting at work...and the next morning they have all reappeared. I have unsubscribed to syncing them, and they STILL come back. It's frustrating to say the least.
I FINALLY GOT RID OF THEM! I TOOK A SCREEN SHOT OF WHAT I DID AND ATTACHED IT FOR YOU. WHEN YOU'RE ON YOUR MESSAGES PAGE, SWIPE THE TOP TO THE LEFT UNTIL YOU SEE LINKS. CLICK ON THAT AND X OUT OF ALL THE MESSAGES YOU SEE THERE. I HOPE THIS WORKS FOR YOU, TOO!!!
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Feb 13 2017 - Galaxy S7. Messages app updated last week and now I too am having this issue; just started over the weekend. I have never had anything like this happen before. I delete texts; they come back. My husband has same phone and setup and he is NOT having this issue. Looks like this issue was known last October...what gives? Is there a fix?