There are no cash surcharges when redeeming BA Avios for travel on AA within the Americas, North and South. That includes travel to Hawaii and the Caribbean.

You do not necessarily have to call BA to book award travel on AA. You can search on ba.com, or on the BA app. But BA sometimes has trouble displaying award availability on AA when connecting flights are involved. And award availability on AA using "partner" currencies (like Avios) seems to have become more difficult to find over the past year or so.

Also, note that BA charges separately for each flight flown, so, if you were to book, say, MSY-DFW-HNL, you would be charged for the MSY-DFW leg and for the DFW-HNL leg.