RohanSix:
There are several issue here but any credit history problems make it very difficult to get a 0% balance transfer card.
What size of fee did you want to avoid paying for your credit rating? Each lender does their own rating however it is well worth obtaining you credit report and checking it for accuracy. You can pay £2 for a statutory paper report, more for an online subscription or "free" if you remember to cancel. Look at Experian/CreditExpert and/or Equifax. Make sure your current residency and electoral roll information is accurate and there is address history for at least 3 years. Make sure you agree with all the account and search information recorded.
An attempt will be made to sell you a "credit score". Up to you but "credit score" takes no account of earnings (mainly account conduct, debt and available credit). Overrated in my opinion.
Don't know whether it's up and running but Barclays have just announced a "no credit search" way of identifying a suitable card product. Might be worth following that up.
At the end of the day, its possible you have been declined on affordability grounds. In that case, while there are low credit limit products available, the best strategy IMHO, is to play a longer game and attempt to pay down one or other of the overdrafts by regular payments that you are comfortable with and revisit the 0% BT in 6 months.