I don't think it's any more complex than running screen on the server, using SSH keys to allow you to connect to the server without a password, and possibly orchestrated with a small script that checks network connectivity and re-establishes your session after it's broken. Your 5 shells are running in one screen session on the server, so you don't have to restart all of them. It's just a simple "screen -D -R". If there is no existing screen session to attach to, you can have screen automatically create your sessions via the config file.

This is pretty much what I do already, except that I don't switch connections as frequently as you. But when I go to a new place, it's just connect with ssh and reattach with screen.