Yeah, this one's common. It's not broken. That weird taste and the black specks (carbon dust) are normal for the first few fills of a brand new filter.
Here's the fix: you need to flush it. Run cold tap water through the new filter for 15 seconds before you even put it in the pitcher. Then, after you install it and fill the pitcher for the first time, discard the first three full pitchers of filtered water. Don't drink it—use it for your plants or something.
If that didn't work, double-check your water source. The Brita filter is not a purifier. Don't use it with water that isn't microbiologically safe. Also, only use cold tap water. Hot water (above 85°F/29°C) can damage the filter.
Heads up: It's normal for a new filter to be wet right out of the package. They're washed during manufacturing so they don't dry out.