Yes, you need to limit the current going thru the input LED of the SSR.
Yes, you need to limit the total current going the 74HC595. That is a logic circuitpart, not intended for driving lots of current - sadly it was selected early on and I'm is working at overstress conditions in many a design.
Use a part intended for high current switching. TPIC6C595 (100mA outputs) or TPIC6B595 (150mA outputs).
Here is a board I offer with 8 SSR channels and TPIC6C595
Can take PVT212, PVT312, PVT412 for switching the outputs.
Youtube videos showing high current (700mA C6060 LED module) being controlled.
