OK, the noise that is causing false inputs would definitely be considered a problem. Its not surprising that the noise may be caused by the current flow to/from the servo motors as these probably take a fair bit of current. Try to analyze where the currents are flowing in a circuit from the power supply to each motor and back again. Then, ask yourself, "does any motor current flow through the ground around my controller logic". If so, then you may have a problem of the two subsystems sharing a common ground too much. In this case, the motor current flow will suffer a voltage drop as it flows through ground, and this potential is impressed across the logic circuits.
A common way to try and deal with such scenarios is called a "star" ground topology, wherein each load that pulls current from the power supply has its own "To" and "from" wiring for power going directly to the lowest impedance point available. All Power+ wiring joings together at only one point and all Power- wiring joins together at one point. So, for example, when a servo pulls current, that current is not shared by the ground wiring of other subsystems, like your controller logic.
Usually, the lowest impedance point is the mains connection to the wall, or in the case of a DC system, the battery terminals.