I want to change my area code, not my number. Is that possible if the number is not being used by someone in that area code?
Basically no.
The number you want in the new area code would have to be an unused number and available for Verizon to assign. This would be highly unlikely.
So how can I find out if my number is available in any of the 3-4 area codes in my current city?
It is not possible.
Different are codes also have different exchanges. It's likely your exchange( that's the first 3 numbers after the area code ) wouldn't in one of the other area codes. And even if it was it's also likely that number is taken.