Pretty much my exact same process. I just don't have as many socks as you do. I end up doing a load of socks about once a week. Have to do one tonight in fact.Chlorine off gases at a certain rate. Whatever chlorine is left after a double wash cycle will off gas very quickly. Letting the socks dry for 8+ hours is enough I let mine dry for at least 3 days before packing them away.
I change the socks every 3-4 days, when im on my last set I wash the others. 3 days later its time for a change I go get the clean ones put the others I'm not using away and change out the used ones and the cycle starts a new. I have 20 7' and 24 4".