Here is a trick to see what the remote control is doing to the keyboard.
On the machine you are remote controlling, go to System Preferences, Keyboard, Input Sources. Put a check mark in the "show input menu in the menu bar."
From the input menu in the menu bar, select "Show Keyboard Viewer"
This puts a virtual keyboard on your host screen. You'll note that when remote controlling your host, GoToMyPC will for whatever reason active the Fn key, and the control key. Sometimes it will unstick by pressing the FN key a bunch of times, but as you mentioned, disconnect/reconnect is the only solution until it starts again.