There is no point whatsoever in hanging on to cash as an "emergency fund". The fact that you have access to a large overdraft and a couple of credit cards gives you a source of funds in an emergency, and gives you even more if you get those paid off. So it always makes more sense to pay off debt than to hold on to money.
Here's what I think you should do.
(1) Pay off that £6500 in its entirety. As soon as you possibly can. Stop using that giant overdraft, but keep it available in case there's a real emergency.
(2) Pay the remaining £500 into the Amex card.
(3) Use your Paypal card to pay off the balance of your Amex card, assuming that £3600 is within your credit limit.
(4) Open a new credit card. Whichever type is the most widely accepted credit card in UK. I'm not in UK so I don't know what this is, but maybe Visa or Mastercard. This will be for your day to day spending. Whenever you want to purchase anything from a merchant who accepts a credit card, and doesn't charge a surcharge for doing so, this is what you'll use. You must make sure that you pay this new credit card off in full by the due date every month, and never use it for a cash advance. That way, you can avoid paying any interest on it whatsoever. And because you get between 3 and 7 weeks of free credit whenever you use this card, this will make it easier to pay off your other debts.
(5) Every time you receive income, figure out how much of it you'll need to retain for expenses on which you can't use a credit card. This might include rent, power, internet, insurance, and a bunch of other things. Pay the difference into your Paypal card. This will enable you to clear the debt on your Paypal card as quickly as possible.
(6) Every time your new credit card falls due, pay off the entire amount, using the Paypal card. If this would make the Paypal card exceed its credit limit, then use the Amex card instead. Never miss the due date on your new card, or pay less than 100% of the balance, otherwise you lose the "interest free" advantage.
(7) Every time you need cash, and you haven't retained enough from your income, use your Paypal card. If this would make the Paypal card exceed its credit limit, then use the Amex card instead. But never use your new credit card for cash advances.
This will be the fastest way to pay off your debts. Once all three credit cards are clear, think again about how much of an emergency fund you really need. Keep whatever credit card(s) you need for this. Destroy the others and close the accounts. Also close the overdraft.