Distributor caps & rotor buttons are cheap & should be regarded as a service consumable, so if you suspect these parts, just replace them. The nice smooth glossy surface inside & out is less condusive to moisture retension [like oxidised paint versus a shiney new wax job]

Somehow, i suspect that the electrics, even if less than 100%, might just be a red herring. Check your carb too as it might just be running lean when cold. Full operating temp often brings the bad running to an end.
You know the usual suspects, warm air to the carb [when fitted] no manifold heat, impared accelerator pump function, faulty choke function. The usual stuff.