You are going to need to add diodes in series with each filament to prevent the current looping back.
The easiest way to do this with your board is to get some very small surface mount diodes and the traces going between the 595 and the filament, soldering the diode across the gap. You should have enough room if you stagger the cuts.
You may be able to squeeze 0603 diodes (which are 0.5mm wide, 1mm long) onto each of those traces.
P.S. That is a very neat and pretty circuit board!