Can I measure the current of a battery by shorting the circuit and by ohms-law divide the voltage of the battery by the current measured to get internal resistance?
I would like to direct you here BU-902: How to Measure Internal Resistance
For a quick idea, measuring open circuit voltage and again with a suitable load (so that battery's internal resistance will be comparable in ration with this load) should give an idea of internal resistance.
With the load, the voltage read will be less than 1.5 V. Say, it is 1.3 V. Then the remaining 0.2 V is dropped somewhere other than the load. The drop will be across internal resistance of the battery.
You know the load resistance value(Rload), supply value (1.5V) and drop across load that is 1.3 V. Internal resistance can be calculated easily.
Please take care of wattage of resistors being used and the allowed safe limit of current of the battery.