More questions than answers here.
Who are ADN? Some kind of phone retailer perhaps? No obvious answer at the top of a Google search.
I've no idea what you are paying 3 mobile £25 + VAT for, that seems to come out of nowhere? It certainly isn't a multiple of £5.67 + VAT.
I'm unclear about why you were getting a cashback of £480 on a contract that was going to cost £132 over 2 years? That makes no sense at all. Why would anyone effectively pay you £348 over two years to take their phone? Was it an incredibly expensive actual handset that you bought up front?
You presumably have a copy of this contract yourself since you say it "clearly stated" that cash back would continue? I don't therefore understand why there's any doubt about whether or not you have a contract with 3 mobile? Is it that they say the terms are different from your copy? In which case surely just sending them a copy of your version would solve the issue?
More info required here I think.