To be honest it does sound a little dodgy from their point of view.
Why oh why would someone claiming benefits as a single person have bills in the name of their non resident partner/boyfriend?
However, as you say he has not been living with you, then unfortunately you will now need to provide evidence. If he has been maintaining a life elsewhere it will be simple to prove. Some excellent ways to do this are -
Electoral roll registration at his own address.
Letters to him re his current benefits showing his own address. (For instance on IS he should have been receiving council tax benefit payable on his own address?)
Ditto bank/building society/credit agreements or other loan accounts with your partner at his own address.
Tennancy agreement & evidence of rent/hb paid for his own address. (Old tennacy agreement is fine of course - so long as he can prove has been there since then iyswim? ie by paying rent for it)
Details of telephone/internet/utilities etc being provided to him at his own address.
Even two or three of these pieces of information will probably be enough to prove he has maintained his own living arrangemenets away from you.
Despite what some folk may tell you I'm afarid that it is now down to you to prove he wasn't living with you. They already have enough evidence to suggest he was, and you will need to demonstrate otherwise.