Hi LakeRat -
Just wanted to say thanks for your help, I solved the problem. 2 DH Support people offered the diagnosis that my sites were collectively using too many resources, stating that WP is a memory hog, and directed me to many, many references on how to optimize a WP website, suggesting that the remedy was to optimize all the installations and then buy VPS if I can’t get the resource use down.
After wading through the optimization literature for a while, it still just didn’t make sense that their diagnosis was correct if I was having the problem on only one of 3 WP installations. So I just set up a new installation at subdomain3.mydomain.com, and everything works fine there. Contact form submissions are still delayed by 60 seconds in the installation at subdomain2.mydomain.com. So it seems like something was just hosed in that installation, and the solution was to trash it and replace it, not spend hours on optimizing and ssh’ing.
Your thoughtful comments helped me remain suspicious of the advice from DH support and saved me lots of time following their fruitless directions. So I wanted to take a moment of that time to thank you! I hope you have a great Thanksgiving.