If the adjuster is in Florida, I am not sure. In South Carolina and the majority of states they only need one party's consent to record phone calls. So if both parties are in SC you only need one parties permission to record a call. However some states require consent of all parties, and if one party is in a state that requires consent, then all parties have to consent regardless of where the recording takes place.
I think you need an attorney from the State were the call was coming from to answer this question.