AFTER MONTHS OF STRUGGLING, I FINALLY DISCOVERED A FIX FOR THIS! TO WHOM IT MAY CONCERN:
To fix this, I followed everyone's instructions and simply tried to just delete the messages, but whenever I reset my phone, the messages would just reappear. I attributed this to the fact that my iCloud was still on. The obvious solution to this seemed to be to just turn off my iCloud backup briefly so the messages I wanted to delete would no longer reappear. I did this but the space taken up by Messages in the Usage section of Settings did not decrease, and the deleted messages ended up reappearing. In a final attempt to resolve this issue, I went into each individual message and used the "Delete All" option. This seemed to be the only way that I could delete messages, gain back the space taken up by the messages, and not restore my phone. Tedious but effective.
Impatient readers skip to here: I turned off my iCloud backup temporarily and individually deleted all messages from WITHIN the message threads themselves. I subsequently turned back on my iCloud backup and all my iPhone problems were solved!