I wouldn't exfoliate for a couple of days after shaving at LEAST because the skin is very raw and sensitive. The shaver takes off some dead skin from the top layer as it is so exfoliating too soon after is too much and can cause a rash/sensitivity. It is a good idea to gently exfoliate beforehand, but not immediately before - say a day or two beforehand.

The itching might actually be lack of moisture? You need to make sure you moisturise immediately after shaving and every day after that.