A B737 departed over maximum gross takeoff weight and landed over maximum landing weight because the center fuel tank with inoperative gauges was not completely defueled prior to adding fuel.

Narrative

During aircraft preflight; both Captain and First Officer discovered Center Tank Fuel Quantity Indicator was decreasing immediately after Captain initiated the indicator test. Both engines and the APU were off. After following up with Maintenance Control; we proceeded with implementing MEL 28-7 which called for pulling the circuit breaker; defueling the center tank; and then refueling it to know the exact quantity. The fuelers loaded an incorrect quantity of gas into the center tank. We took off assuming we had the gas written on the fuel slip/loadsheet. We thought we had 8.3 thousand but turned out to have 16.5 thousand with an additional 20 thousand in wing tanks. The true quantity of gas was only ascertained in flight when wing tanks started burning down. We then realized we took off above max takeoff weight and had to land about 6 thousand above max landing weight (after coordination with dispatch). The Captain monitored the defueling operation and most of the re-fueling. The First Officer was in the flight deck monitoring the low pressure lights. Then the pilots swapped places. Both pilots verified the gallons on the fuel truck matched the pounds we needed in the center tank. While the correct fuel quantity was likely added; the pre-existing fuel in the center tank may not have been de-fueled correctly. Still need to discover the root cause. If it is defueling operations; then review best practices for pilots to monitor the fuelers.