As you described the problem, that's most commonly associated with a battery that cannot hold the load. I assume you properly said that the battery was changed, not charged? If it was charged, make sure they did a load test on it (some places don't...and say they did...). If it's a new battery....then I'd suggest your starter is having issues. The click is the starter solenoid, I'm not sure what the buzzing is, but I'm pretty sure it's related to your starter.
Go figure.
I would guess there's a possibility you've got some not-so-good contacts associated with your starter, which could definitely be aggravated by damp weather.
b