How to shift resistance range of Linear Potentiometers by X ohms in the Positive or Negative direction

Given that you've said R7 is set to 0 ohms, the two fixed terminals of R6 are shorted together. When the wiper is at either extreme, its resistance to either fixed terminal will also be 0 ohms. When the resistance is 50%, both halves will be in parallel. Given that this appears to be a 50 k pot, we'd expect

$$R_{eq}=\frac{R_{max}}{2}||\frac{R_{max}}{2}=\frac{R_{max}}{4}=\frac{50000}{4}=12500$$

just as you measured.

Apparently you hook up the potentimeter as a proper potentiometer, that is: the wiper supplies a fraction of the toptal voltage to the A/D ipnput. That fraction does not depend on the Ohmic value of the potentiometer: when you tae a higher value, then for a given position of the knob bot 'sides' will get higher values, but the ratio will not change.

Note 1: the voltage supplied by the wiper will not change when you take another Ohmic value for the potentiometer, but the impedance (effective series resistance) does change. As long as this impedance (worst case 1/4 of the value of the potentiometer) is lower than what the A/D input specifies as maximum this is not harmfull.

Note 2: the most common potentiometers are linear, but there are also logarithmic (and anti-logarithmic) potentiometers, which will (for a given position) give a very different ratio.