Coils can definitely fail when they get hot and work again after cooling off. That's a pretty common symptom before they crap out entirely. If yours gets + power from the choke (as they are sometimes wired) that might explain the discoloration on your carb.
There's a bench test you can do on the coil to check internal resistance. The specific ohm specs are in the service manual if it's stock, and online if it's an aftermarket coil. That and checking to see how it is wired (and the condition of the wiring) would be a good starting point. Coils are cheap enough that it would be worth checking the wiring, replacing the coil, and seeing what happens. Worst case you end up with a spare.
"Speed's fine, but accuracy is final."
- Bill Jordan
1998 Donzi 21LXR
1971 Boston Whaler 13