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@allObjects
Not much a surprise here: the cells have a 225mAh capacity, which roughly says: you can pull 22.5mA at nominal voltage - 2x3V when serial - for about 10 hours before the voltage drops below usable... Your relay - according to datasheet - is guzzling about 90mA... which is four times what the cells can push with keeping up the nominal voltage...
Running relays in a battery driven world is a challenge... I do not know what you want to switch in the end, but if you driver to switch that is that draining, it's going not for a 'long' time...
How often for how long over what time should the relays be on? You have to look for an alternative... If you still think you need a relays, you could think of a bistable relays that sucks not more than 20mA. Since such a relays needs only a pulse to change state, a capacitor can help to support the spike in current consummation.