Electrical – How does temperature compensation work for strain gauge in wheatstone bridge

You may be seeing thermal EMFs, as well as thermal drift of the resistors.

Blowing hot air with a hot air gun is not the way to test a circuit like that. Use a proper environmental chamber with a fan inside and control the air flow over the board (maybe with a bit of foam). You may have to wait an hour or two for the temperature to stabilize. The hot air gun will lead to wrong conclusions. You may see dynamic changes that even out as the gradients disappear.

Do not have anything on the board that dissipates a lot of power (I assume you are doing this).

Leaded resistors can be a bit less prone to change from external mechanical forces. Try flexing the board a bit and see if your reading changes. Chances are this is not your problem, but it's worth a check.

The way I do it is to use an instrumentation amplifier (TI, Analog devices etc.) that amplifies the difference signal and then apply a variable voltage to the reference input of the InAmp to balance the output to zero.

An InAmp basically comprises three op-amps as shown below: -

Note the "reference input" bottom right and the potential divider that sets the offset voltage on the InAmp output. This can be a potentiometer or even the output from a DAC. It all depends on what you want to do and how automated you want the zeroing process to be.

Some applications will allow a slow integrator to be fed from the InAmp output to the reference input and in this way you are always auto zeroing but at such a slow rate that the measurements you make are largely unaffected. This may or may not be of interest to you of course.

There may be a way of adjusting the HX711 (load cell amplifier) to do what I suggest above so if you want help in that direction, please link to a proper data sheet for the chip.