The good news is that the fact that you are doing push ups and even considered attending OCS with this injury makes it sound like it is minor and the treatment is going to be rest and advil. Without insurance the doc may not even bother sending you for an MRI, since without surgery the treatment for a sprained rotator cuff or AC joint is the same.
Glad to hear this. Jumping the gun is in the realm of possibilities for me... I panicked into a doctor's office last summer with a "sports hernia" that turned out to be an abdominal strain. Luckily the ortho I'm seeing is top-notch and a family friend from little league baseball days.