Cottage Rental Agreement

Our winter tenants are willing to send a deposit and sign the booking form. However, since they are not in our region, we need to do this in the mail. I agree, but have not yet signed. Should I send them unsigned, sign and send copies, or do I sign, then mail and having to send them to me and I send copies?
I think they should sign first because if I send a signed, it could change.
Thank you!

The right way to do is send them unsigned. You are absolutely right, if you send them signed and they make changes, you can not prove anything.