You know that the fuel and temperature gauges are related, right?

Both gauges are essentially measuring current from a common, and constant(!) voltage source that is grounded at the sensors, the amperage determined by the resistance at each sensor. If the voltage (let's call it a reference voltage) fed into the sensors' circuits is wrong, so will be the readings of each gauge.

Because the voltage in the car's 12 volt system varies as the battery's charge, load of accessories, alternator's output, and general condition also vary, the reference voltage is the unvarying output of a voltage regulator chip (a solid state, 3-terminal IC mounted on the back of the instrument panel circuit board), arbitrarily selected as 10.0 volts because it has to be below the range of the car's system's voltages in order to perform reliably.

As noted above, if the chip's output is faulty, so will both gauges' readings.

I'm not saying that this chip is your problem, but having noted, "...the fuel gauge was also off: with tank full, needle was at 3/4; when at E dead bottom, there were 5 gallons in the tank...," it is worth considering.