Power consumption when DC motor runs on battery
Before calculations you need to be clear about some fundamental concepts about motors.
 "It said that current required by motor = 3 × current required while running on starting."
This statement can not be true, because motor draws more current at start up than it rotates in operational angular velocity.

"When we run the motor on battery eventually battery voltage got dropped, taking more current."

This statement is also false. Motor speed ----> Voltage, Motor torque -----> current.

My advice is as follows. Your application does not require detailed calculations with motor specs. What you need to do is measure the current when your motor rotates under the load, which you choose. Then compare it with battery capacity.

About voltage drop of battery, it does not draw more current when battery gets low on Voltage. Voltage is about speed(Theoretically). Thus, as long as your load is the same, theoretically, motor draws same current. 

NOTE: Of course current change when the Armature voltage changes, but this is negligible.