How to Calculate Op-Amp Level Shift for differential amp and gain of 2
It is true that shifting the gain will change the DC offset of the circuit above but this can be calculated, so that if you do change the gain above, you can also change the resistors R2 and Rg to keep the offset the same.

Can&#39;t balance differential amplifier circuit
That means if you change Rf, then when selecting resistors R2 and Rg the fraction
\$\frac{(R_f+R_1)*R_g}{(R_g+R_2)*R_1}\$
needs to stay the same, and then the output offset from V2 will also stay the same.
Another circuit you might want to consider is an instrumentation amplifier, which changing the gain does not change the output.