@SarahGen-- Keep applying ice and keep your wrist elevated. If you keep it at or above your heart, this will prevent your wrist from swelling more.
Swelling is a sign of injury. It could be a sprain, but it could also be something more serious like a fracture or a dislocation. I think yours might be just a sprain because a fracture or a dislocation is usually extremely painful. You also cannot move your wrist at all in these situations and the swelling in the wrist will be very serious.
Keep your wrist immobilized for now, but it's a good idea to see a doctor for a physical examination and an x-ray. Especially if it's dislocated, it needs to be realigned within six hours. So you don't want to waste time.