Take a look here: http://nicholasclewis.com/projects/inkshield/theory/
Your problem is probably that you wired them the wrong way around. The big contact (com1 and com2) is the source, the 20V, not the ground. All other contacts are sinking. ULN2803A com should also be connected to the 20V, and the GND to ground. ULN2803A’s are sinking transistor arrays, not sourcing. Then wire all remaining contacts to the outputs of the ULN2803A’s and your signal wires to the Inputs of the ULN’s.
This should fix it.