With exchanging details in the case of an accident as you mentioned, if you not comfortable with that maybe you could consider ways you could prepare, for instance you could make sure you have all relevant details already written down on a piece of notepaper which you could hand to an other driver. Maybe on another piece of paper you might keep in your wallet you could write down procedures you should take in the situation, for instance what details to take, photos, call the cops if needed, ect.
I know what you mean when I get nervous or feel pressured I sometimes also forget half the things im supposed to say, so sometimes I write it down and go off my list.