If your coil is that hot, something is definitely wrong with it. Either it's not properly ballasted or has some kind of internal short circuit. In any case, you are pulling way too much current through it. This current is also going through the points, and you are probably burning the points as well.
To confirm the obvious, disconnect the coil and measure the resistance with an ohmmeter, between the two low-voltage terminals. (If you don't have one, you can get a perfectly good one for under $10 from Harbor Freight.) If it's internally ballasted and not defective, you should have 3-4 ohms; if not, you will see 1-2 ohms. (Before making the measurement, short the ohmmeter's leads together and see what it reads. Probably about half an ohm. That is the meter's offset error; subtract it from what you measure at the coil.) You can try adding about 1.5-2 ohms of ballast; say, a ballast resistor from a parts store, or replace the coil with one that is internally ballasted. It's a safe bet you don't have any external ballast in the circuit.
If adding ballast doesn't fix it, the coil might simply be fried, and you'll have to replace it.
Then, check the points and clean them. They almost certainly will be somewhat burned. Even if you plan to replace them, it might be good just to clean them for now, with an ignition file, and replace them when all is running well. You don't want to risk frying a second set.
While you have the digital multimeter out, check the system voltage. I doubt it will be way out of order (i.e, more than about 15 volts) and even if it were a little high, it wouldn't cause what you're seeing. If it really is high, the regulator is bad (but your generator is doing great!)
This may not be the only problem. (Sorry to be so optimistic!) It sounds to me like there might be a fuel system problem too, but one thing at a time. This one should be easy to fix. Good luck!