You do not have to exchange US$ for BZ$. US$ are accepted everywhere and you are given change in BZ$. I always carry as many US$ 1,5 and 20's as I can. Save spending your $50's & $100's for large meals, tours, rentals, etc. If you don't watch it you will end up with a lot of BZ$ left over and they are only good in Belize. If that happens, what the heck, it just gives you an excuse to go back again and spend your money!!! Most debit and credit cards have international transactions fees and can run as high as 5%, so check with your bank. Always good to have with you in case of emergencies.
I don't know where you are staying, but many places have safes in the room or available at the front desk, so lock it up and only carry with you what you need for the day.
Enjoy (I know you will)and you will start planning your next trip after a few days on the island!!!!!!