What causes hand swelling, along with pain moving my wrist?

Carpel tunnel: There are many reasons to have the symptoms you described, a common cause is carpel tunnel syndrome due to swelling of tissues at the wrist. It is a treatable entity and should be excluded before considering other possibilities. Your physician may do tests to validate the suspicion.

Wrist sprain?: Hard to say. A wrist xray would be a good place to start to make sure that there are no fractures or arthritis. A sprain is possible if you had a previous trauma. Wrists are slow to heal as we are always using them, similar to the ankle.