How does inductance of armature winding response to changing load in motor?
This increase in current when you load a motor more is not related to inductance. It is mostly because of the mechanism of conversion of electric energy to mechanical energy, which follows Lenz's laws (among others). There is no change in inductance happening.

When you increase the load on a motor, you are in effect putting more load torque on the motor, which opposes the torque produced by the motor. This motor-produced torque is directly proportional to the current flowing in it. You applying the load reduces the speed and draws more current. It is like you lifting a 5 kg load up a hill and then someone puts another 5 kg weight on you. You would need to supply more of your muscular energy to maintain the same speed, just like the motor draws more current. 

The inductance present in the coils is due to the leakage inductance, which itself is present due to the imperfect (rather incomplete) coupling of the magnetic fields of the rotor magnets and the stator coils. This remains constant and is largely fixed by motor construction.