If insurance and a doctor's visit aren't an option, then here's a practical paramedic's best guess: it does indeed sound like a sprain or tendonitis of some kind. The hand has amazingly complex nerve/tendon and muscular structures.

It's true that a doc wouldn't be able to do much other than rule out a break with an X-ray. You could have a fracture of a small bone. But if it's the more likely possibility of a sprain or tendonitis, then remember RICE: rest, ice, compression, elevation. The compression (like with an Ace wrap) and elevation are particularly important if you have any swelling. OTC anti-imflammatories like Advil and Motrin are great for sprains or tendonitis. And use that hand as little as you possibly can for the next few weeks. If you coudl immobilize it in a sling, even better. Both sprains and tendonitis take lots of time to heal. Time and the RICE approach are what anybody trained in practical or orthopaedic medicine would tell you.