Hi Matthew. Are you running 16.3.6754? I did some testing on our latest public release, and I'm not able to replicate that behavior. On an email with 10 replies on the email chain, the signature is added on the user's reply every time. It's never added to the boxes that display the previous messages or at the very bottom of the email chain.
I tried doing a Reply vs Reply All, adding the signature right away vs after typing/deleting some text. I even added my reply in the "boxes" for previous replies, but the signature is always added where it should be.
Are you able to determine any specific replication steps? For your account and your user's... Does your signature default to None, or does this happen when changing the Signature / From address?
I'm hoping this is something I can replicate and get over to the development team for their correction. If I can't replicate the issue; however, I'd like to get you in touch with the Support Department for their review.