Aha! I figured out the cause of my server slowdown.
Every webhost assigns domain names in your accout to an actual system user – and each user can use a certain amount of resources on the server. Currently, all of my blogs are assigned to the same system user. So when all my blogs get trafficked along with everything else I’m hosting, things get delayed or result in errors.
To solve this problem: I have to move my blogs to different system users. Sounds a bit daunting but hopefully it won’t hurt too much.