Good, the display is working properly and by shorting the input it suggests the measurement is also working. The next thing to check is the resistor values you used. You don't show any values on the schematic but I suspect you have copied them from the 200mV FSD schematic in the data sheet. This means the maximum voltage you can measure is 0.199V so applying 1.5V would take it well over range.
The other thing to watch is the power supply, the display will take quite a lot of current so as the segments turn on and off the drain on the power supply will change. If you do not fit decoupling capacitors across the regulator input and between the V+ and V- to ground you almost certainly will have instability on the supply rails and this will result in random digits. I suggest you fit a 10uf and 100nF capacitor in parallel across each of those points to keep it stable.
See if that fixes the problem.
Brian.