Entering the U.S. with over $10000 in cash (or equivalent in any combination of foreign currency) will require you to fill declare the cash and will raise a few red flags. They may require proof of where you earned the money and ask why you're choosing to bring it to the U.S.
Typically, withdrawing money from an ATM is an even better option for getting local currency than exchanging it at your bank. Simply use a U.S. ATM with your UK bank card to remove funds from your account in dollars. This will sometimes incur a fee from your bank, but it is possible that your bank will reimburse you for these fees. Check your bank's website for more details.
Since the U.S. Dollar and the Pound are both major world currencies, foreign exchange spreads should be very narrow. Be sure to compare rates. Generally, you will get the worst rates at the airport and the best rates by withdrawing at an ATM.
Consider also using your credit card. Some cards have an overseas transaction fee, but if yours doesn't, you can bring less cash by using your card for most purchases.