It seems like you may have some connection issue somewhere. Were there any connectors that you touched during your carb rebuild? If so, start there.
If you don't get 12v on your solenoid while cranking, that's not right. But, it could be caused by various reasons. One of which could be the ignition switch, but not the only possible cause. There are a lot of other connections in that circuit.
When trying to get my 79 going, I went through and cleaned all important ignition connections, including battery posts, frame and engine grounds, and all relevant ignition circuit connections (coil, igniter, starter, cold start circuit from starter to coil +).
About the variations in voltage on your coil + terminal, 12v sometimes and 7v sometimes is expected on a system that uses a resistor circuit to feed the coil. When current is flowing through that resistor (or resistor wire) only 7v will be delivered to the coil +, so that it doesn't burn coil, points or igniter.
But for starting, a full 12v is needed to make starting easier. On the 79, that 12v comes from the starter when it's cranking via a jumper wire.
Your model may be different, but it sounds like you do have a resistor in the ignition circuit. Check your coil + when cranking and see if you have 12v.
Still, I would go through and clean all your connections. Time and corrosion can create weird issues with your electrical system.
Then start checking voltages again.