You shouldn't ever use a bare 595.
Be sure to have some capacitors handy (100 nF is widely used for this).
Put them across all power lines of each chip you're using.
Read datasheets of chips before using them.
Your body interferes with the interference, so you'll see some interaction/
LEDs especially need good decoupling since they carry much larger currents than
most logic level signals but switch just as fast - suggest 0.1uF for every chip, 10uF
for each board, minimum.