Been in the Caribbean often. Most of the large shops accept US dollars and give change in same. Small shops will give local currency so take plenty of 1 and 5 dollar bills ashore.
Everyone will take your US$
Everywhere we have visited took Us dollar and change always came in US dollar. Islands included Barbados, Tortola, Antigua and St Luci.
All islands accept the $. Even British Virgin Islands! some islands, like Antigua only give you loose change in their local currency, east caribbean dollars. Aruba will give you florins. Cayman Islands in their C.I. dollars. All of this is just bits of cash to give to the airline on your return!
When you pay with us$ ask for change in us currency, no problem on any of the carribean islands i've been to.