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I'd try the wiring kit first. Even if its not the culprit, you still need it. I had that problem for a little while and it just turned out to be a bad connection
I have the DDM kit and I only had this problem when I used the harness kit. When I plugged the kit directly to the bulb sockets, the problem went away. So the harness will make the issue worse if anything I'm thinking.
Try turning on the lights AFTER you start the car up and you shouldn't have a problem with only one side coming on. If it still does, then it's your ballast, but they should replace that for free since it has a lifetime warranty.
Doesn't DDM have tech support call them and up and get the real solution. If you ask me I think you would need a canbus system or a relay harness. Not much to an HID system.