Starts fine when it's hot and/or dry, but when raining or cool, it just cranks.
I saw some of the previous questions on this same issue, but have been unable to determine what the issue is.
I replaced the plugs and wires and removed the distributor cap. Everything inside the cap looked dry and very clean. I sprayed WD-40 inside, wiped, applied grease to the seal and replaced the cap.
Then I noticed that when I touch the center of the cap when the car is running, I feel a shock. Does this mean that the cap is losing charge and needs to be replaced or would you recommend replacing the entire distributor assembly? I have about 125000 miles on it.
Thanks,

I would replace the distributor cap and O-ring immediately. Hopefully, the stray secondary voltage has not damage the ICM, coil, and CKP/TDC/CYL sensors. The link below should help:

https://www.askmehelpdesk.com/cars-trucks/faq-how-troubleshoot-repair-maintain-hondas-46563-5.html#post520809

Because of the age and mileage on your Civic, I would also replace the Ignition Control Module, rotor, and coil.