Dear Emily,
Thank you for visiting the Family Practice Forum.
I am going to assume that your doctor read the Xray correctly and therefore you have no fracture. It sounds like you have a fairly significant sprain of the thumb, however. This would typically improve with the use of "gentle" range of motion exercises and possibly the use of an anti-inflammatory medication such as Motrin. I rarely, if ever, put patients with a thumb sprain in a splint, unless they continue to "overuse" the sprained area. Sometimes a splint helps to serve as a reminder to "rest" the area. It sounds like you have probably not rested the area long enough.
If the pain persists despite rest, anti-inflammatory medication (i.e. Motrin), and gentle range of motion exercises, after a few weeks your doctor may consider sending you to a "hand therapist" (a type of "occupational therapist") for some more aggressive therapy to the thumb.
Remember, in many cases sprains may take up to 4 weeks (or more) to fully heal. The most important factor in helping a sprain heal is to make sure you "REST" it. If you are going to wear a splint, make sure you take it off at least twice a day to perform gentle range of motion exercises. This will help avoid stiffness in the joint.
I hope this helps!