You've figured it out more or less correctly. dimissuram agrees with se, which is accusative (because it's the subject of the future infinitive dimissuram esse) and feminine (because the antecedent is Dido). It's reported speech after promisit.
But dimissuram esse doesn't necessarily mean "about to send the Trojans away". It's just a (periphrastic) future infinitive. "She promised that she would send the Trojans away safely." The future participle alone, when not part of an infinitive, can usually be translated "about to." (This is an instance where a Latin adjective tutos is best translated as an adverb "safely".)
laturam is also a future infinitive agreeing with se. esse does double duty for both dimissuram and laturam. (esse could be omitted and merely understood, and probably would be in "real" Latin.)