"ABSOLUTE MAXIMUM RATINGS (1)
Continuous output current +/-35mA
Continuous current through VCC or GND +/- 70mA
(1) Stresses beyond those listed under absolute maximum ratings may cause permanent damage to the device. These are stress ratingsonly, and functional operation of the device at these or any other conditions beyond those indicated under recommended operating conditions is not implied. Exposure to absolute-maximum-rated conditions for extended periods may affect device reliability."
So 35mA rating is not recommended long term.
And the power supply pin only supports 2 pins at that current.
Further design questions:
Do you have 0.1uF cap on the Vcc pin of each shift register?
Are you sinking current (low outputs) to turn things on? I would use TPIC6C595 (100mA per output) or TPUC6B595 (150mA per output).