My 04 Accord cranks slow when it is damp and after about 12 seconds of cranking, the battery is (or appears) dead, and I just get the "clicking" (like a dead battery).

1) The battery is new. I had Auto Zone test it and they said it is working fine.
(terminals are spotless).
2)AZ also said alternator is excellent, and I see that it puts out about 14.4 v.

It always starts great when dry, but if I let it sit "for a day" in damp weather, it will crank really slow and dead. If I put on battery charger for a half hour (on 8 amps), it will then start again.

If it were the starter, I do not see why the battery would die after about 10 secs, and then I get the clicking (like a dead battery).

Any advice would be greatly appreciated!