When you say cash, do most places accept US dollars or is most everything in BZ?
You can use either everywhere but will normally receive your change in BZD. Just make sure to use up your BZD before you leave the country.
As for credit cards, Capital One doesn't charge a foreign transaction fee. If you have that one or another that doesn't charge, you should charge the more expensive parts of your trip (resort plus any tours booked through your resort, golf cart rental). Nicer restaurants will take CC and won't charge an extra premium. Now that we're living here, we use cash for nearly everything, but it's a bit more challenging on a trip. Many places take traveler's checks as well, and you can cash them at the bank. Kind of a pain, but that's what we did when traveling here to avoid carrying tons of cash.
ATMs are available, but our bank charged a foreign transaction fee, plus there was a local fee for the ATM as well. We had a lot of unexpected surprises w/fees after our first trip here so wised up for subsequent trips.
Good luck and enjoy your trip!