I would try Mike's idea first, but if it doesn't work here are a few more solutions:
#1. Try adding a 1K resistor to ground on each of the inputs to the darlington arrays (ULN2003A pins 1-7). That is an attempt reduce the leakage current by forcing the transistors completely off rather than mostly off due to leakage in the 74C164. You can try just a few to start out with, but it will only affect the circuits that have the resistor.
#2. If that doesn't work add a resistor on each of the outputs (ULN2003 pins 10-16) to +12, 1/4 watt 2K should be fine, but experiment if need be. These resistors will get warm when the corresponding bulb is on. A lower value will waste more power but be more likely to fix the issue. If you go below 0.7K you should look to 1/2 watt instead.
#3. Of course, you could always replace the darlington arrays with discrete mosfets and be done with it, but that's quite a bit more work, and you may still need the pull down resistor from #1. This is more of a nuclear option rather than just a simple circuit modification. If you decide on this route it will probably be easiest to place the resistors and mosfets on a proto board and jumper to the main circuit after removing the ULN2003A chips.
Hope this helps.
Marc