Maximum Number of Zones

[astrogerard]

Connecting GND to the hc595 should be as short as possible and the best is connecting all hc595 to a common GND.  You could use the bottom rail for it as long as there are no ground loops and everything is decoupled. For decoupling you could connect the top 5v also to the bottom rail and put 2 caps (0.1 + 47uF) on it.

Pin13 to GND is safe to do and you eliminate a (long) wire.

Instead of the 4k7 as a pull-up you could also use a 10k. If pull-up is part of the issue even with 10k you should notice the difference.
edit: since the Pi's pins are configured as output it shouldn't make any difference. You could try it with 10K anyway

If you don't have a 100 resistor you could also use a small coil if you have one.  May be you have some scrap pcb's and could salvage components from there. I have many components in stock but quit often I miss a certain value. In that case I browse through a box with old circuit boards and mostly will find something close to what I'm looking for. May be this works for you too.

Another experiment you could try is connecting the hc595 to +3.3V instead of +5V.  The 3.3V on the Pi is way more stabilized than the +5V so if the issue is related to the +5V supply you should notice quit a difference on 3.3v 

You will get there eventually. :-)

Edit: pull-up