You've got a lot going on there.
The bulb you are using is obviously not a sealed beam, and frankly, I've never seen that type of connector setup. Since you must have taken conventional sealed beams out, I'd hook it all back up with them and get the bugs out of the relay harness first.
So, I would unplug the headlight doors at the motor and try the harness with the old bulbs first. Then figure out what works and what doesn't work.
It wouldn't surprise me if you have a bad relay. I'm not a fan of the clone relays that the Putco harness has, so I'm always suspect of them. I know some of the guys here have had good luck with them, but I always push to use original Bosch/Tyco relays.
Concerning the bulbs, looking at the Hella website, their sealed beam conversion bulbs don't look like this. What exactly are they?
My Hella Lights :: Vision Plus 5¾" Conversion Headlamp