A motor generates back EMF dependent on it's speed. If speed is zero, no back EMF, so it is only the DC resistance that limits the current. Free running speed generates slightly less back EMF than the voltage the motor is rated for, so you get the input voltage, less the back EMF at that speed, divided by the DC resistance, gives you the free running current. If you slow the motor, the back EMF drops, giving you slightly more of a difference between EMF and back EMF, so the current goes up.
Battery voltage drops because they have their own internal resistance. As the current goes up, you lose more voltage across that resistance. No device is purely a battery, resistor, capacitor or inductor. A battery has capacitance, series resistance, inductance, just as anything else has, and the parasitic bits often aren't even linear. One example is capacitors that are used for smoothing rectified power. They have low ESR, which is Effective-Series-Resistance.