Strategy to balance a Wheatstone bridge for strain measurement
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.