According to the Dreamhsot Newsletter they are about 1/3 of the way through the sarge upgrades... So given that it took a bit less than a month to accomplish this, then in a worst case scenario it will be two more months before they are finished.
However, the server upgrades shouldn't be making Apache or Mysql respond slower. I'm wondering if the problem really is that someone is overloading one of the servers you're using - and if that's the case then you would want to contact support to ask what's going on.
I can't say that I do anything all that intensive with my sites, but they have not been responding slowly.