How can I get the ball rolling on my DD 368?

Go ahead. Call Brigade. That may not end well for you as you could very well be asked "Did you run this thru your Chain, first?" You have to realize that there is no set time table to have 368s done. The AVERAGE time is 3-5 months (Command depending). Not to mention the fact that there is nothing stating that they have to approve it. You also need to remember that it is sitting on the desk of a Full Bird Colonel. A Colonel that has many other tasks and responsibilities to accomplish. Do not, I repeat DO NOT, rush a Colonel. Then, after that, the 368 still needs to go higher to TAG.

OK. Your DD368 probably isn't sitting at Brigade since they aren't the approving authority, unless you are one of those odd separate brigades that is commanded by a general. It's probably sitting at your State Headquarters waiting to put it in front of the General. Since most of this kind of activity happens on drill weekends, figure that it will take a few months before it hits the level that can approve it. I would consider six months fast from what I've seen in the past.
Now the bad news. The General does not have to approve it and a lot of states are pretty strict about releasing someone that isn't close to finishing his original contract. I would check with my Readiness NCO first about where it's at. You can request an open door with the Commander and see if he would let you hand carry it. None of this is mandatory for them and they can just tell you no.

You can use your leadership to make an inquiry but I don't know if I would recommend a PFC calling Brigade directly to rattle some pots and pans