If you want to stick with your original circuit, there are a couple of small changes that will improve its performance. It has two balanced bridge circuits in series, and the important thing to know when you are not chasing millivolts or microvolts of precision is that it is the balance, or ratio, of the two sides that matter, not the absolute values of the components.
For the sensor bridge, reduce R10 and R11 to 15 Ohms each and insert a 10 ohm pot between them, with the low end of the bridge taken off the wiper. This will let you balance out any tolerance errors in all 4 fixed resistors with one adjustment. You could just replace R10 and R11 with a 50 ohm pot, but this way gives you better adjustability.
For the diff amp (which is a balanced bridge circuit with gain), again it is the ratios that matter most. If R15/R12 = R14/R13 you will get maximum common mode cancellation. Having perfectly matched absolute values improves the differential input impedance, reduces input bias current errors, and gives you predictable and repeatable gain, but again that is the stuff of higher precision needs. Inserting a 100 Ohm pot in between either resistor pair (but not both) with the opamp pin connected to the wiper will give you a single point balance adjustment.
If cost and/or size is important, put all of the component tolerance corrections in one basket: lose the bridge pot and keep the diff amp pot. Not preferred, but better than nuttin.
To adjust the bridges, first dead-short the R8-R10 node to the R9-R11 node, disconnect the oscillator, and tweak the diff amp pot for 0 Vdc at pin 7 (measured wrt the circuit virtual ground). you should be able to swing the output above and below 0 with the pot. Reconnect the oscillator and retweak for minimum bleed-through (there shouldn't be any, but good to check). Next, remove the shorts and adjust the bridge pot for minimum signal output at pin 7.
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