A tenancy agreement is certainly reasonable proof, although do you not have an up to date driving licence you can send, or any bill such as council tax that you can send? You probably need to explain very clearly why their searches (presumably credit searches) show you moving TO an address on the day you moved to a different one.
I think they are being exceptionally patient with you, and you do run the risk of them, rightly or not, thinking that you are not being completely truthful, so it is in your interest to explain as clearly as possible what has gone on with your address, and why you think it affects the amount you should reasonably pay to resolve the situation you are in.