Actually it would be Dido looking this way and that, because the participle volvens is nominative there and would agree with the subject of the sentence (if it were Aeneas looking, it would have to be volventem). It wouldn’t necessarily mean that she avoided his eyes, merely that she was looking all over his body.
Incidentally, in Latin literature the accusative of Aeneas is usually given in its Greek form Aenean.