to answer your question, yes height does matter. The higher the tank is, the more pressure on the needle valve.
That said, if its only an additional foot or so, it should not make a difference. The fuel pump can develop 1 to 2 psi, itself.
I don't have a parts book close-is this the carburetor with the plastic seat and float assembly in it? If so, renew the little o-ring around the seat/between the seat assembly and the carburetor body. If that o-ring is cracked, gas completely bypasses the seat.
If the seat is in the carburetor body itself, lap the seat using a Q-tip. I wouldn't use the new needle because you may damage it and that would compound your original problem.
Also, some of those carburetors are available aftermarket very reasonable priced.