I was wrong in that old forests are net carbon emitters. They are still net carbon absorbers, but not as much as young forests. The reason for this is that trees also respirate, taking in O2 and emitting CO2, and this balances the CO2 absorption of photosynthesis. Old forests are still net carbon absorbers because trees have to repair themselves, and that matter still comes from the air. But intuitively, the amount of carbon absorption from maintenance vs. the amount from growing larger is much less for the former.

When they rot the carbon mostly goes back in the air.