Has anyone had success contacting a unit directly to be a part of that organization?

No but I made it mandatory to designate a sponsor for all the soldiers and NCOs coming to my unit.

Yes, of course it depends on what level your orders have disclosed, Battalion vs. BDE. You can reach out and as who your sponsor is. In fact you should be able to go to their web page. Thank you for your service.

Yes but only if you're on assignment to that brigade and you want that particular battalion.
It's a common misconception that you can just contact the CSM in the unit that you want to go to and they will be able to by name request you. That's not how shortages, strengths, or priorities work. If that person works in a nominative assignment, or the school house, or some position that has an influence over your branch manager, that is a possibility. If you're just hoping to go to a BCT or any other conventional unit, the CSM doesn't have the power to over rule the HRC manning guidance which has been approved and authorized by the Commanding General of HRC.