Do you know what is the stall current of each servo and what is the total stall current for all servos combined?
It is likely that the servos will draw a lot of current until they settle at their stable positions - and they all do this at the same time at power-on.
I wonder if this is a case where a current limiting power supply is too clever for the application. Something that could supply more amps for a short period (a few seconds) may be better.
...R