US$ are welcome everywhere in the Caribbean.
However the Euro seems to have 'gone up' in relation to the US$, so might be a good idea to keep your Euros as Euros, and then just change them in a bank out here for local currency as and when required?
The Barbados and Eastern Caribbean dollars are both linked to the US$, so we really notice it if we travel to Europe, or even just when buying anything in GBP or Euros.