I don't know what's normal for a voltage drop when the system is powered on but 1.9v to .8v does not sound good.
The 1.9v line should have a low resistance to ground (~20-30 ohms) because there are some low resistance capacitors on that line, right under the GPU. You just never want it to be 0 ohms.
When a working GameCube stops working only after you've wired something, it's probably because of... well, the wiring. It doesn't seem logical that anything is suddenly going wrong unless you have some serious static electricity on you that's killing the boards when you touch them.
How thick are the wires you're using now? I don't know why it never occurred to me to ask. With an issue like this you should try to thicken the ground wires first (or just add more). If that doesn't provide results, try the main voltage wires.