I can't straighten out my thumb at all and it's very painful. Should I get a cast?

No. Could be a trigger thumb. The tendon that flexes the thumb can become swollen and get caught in a tunnel that it must slide through. This can be very painful. You can try a splint and advil or aleve, (naproxen) but these do not usually work. A cortisone type injection into the thumb can provide complete relief. See a hand surgeon to get it checked out.