the air pressure sensing is pretty easy. the sensor itself needs to be in the same chamber as one of the two pressures.
if that means in an enclosure that has a tubing connection, then that should work. if it means installing it inside of the device that has a pressure, that should work too. all pressure sensors reference two pressures.
the teesny has a 13bit ADC (useable) device. that will give you a much higher reading.
you did not state the actual process you want to measure. check out the device specs for error.
accuracy is 1-1/2% for your device. most digital will then stipulate device error plus/minus 1 digital step.
the 10 bit will already be more accurate than the device. if you are looking for high sensitivity, then get a device with the range you want and that as an accuracy in the range you want.