I have seen that a new cell is 1.5V but it falls rapidly to 1.2~1.3 V and stabilises there.

Erik, that is not the property of his batteries but a quick discharge. I assume that he is using the normal disposable type battery cells. These type of batteries are considered useless when they reach at 1.2-1.3V.

The real problem seems to be with his circuit which draws 270 mA. I can't comment on this since I don't know anything about his circuit.

There is no use of using 5 discharged batteries instead of 4 good ones, to get 6V.