Based on my experience, I can say that stirring does not affect the brewing process in a French press a lot. The initial stirring is just for homogenously distributing the grounds in the water. Then, the rest of brewing happens itself in the water by chemical extraction itself and the main factors are time and temperature.
According to your question, I can say that you may be over extracting if you are waiting for the water to cool down.
One quick recipe and other related discussion are mentioned herehere.
