Hello Good Evening,
You either have a tendon injury
of any metacarpal bone, but in my opinion you might have a soft tissue injury
which can cause severe pain
because of compact space at that site between tendons and bones.
Swelling can not cause a fracture but fracture can cause swelling frequently but not always, like in hair line fracture and of compact site like dorsum of foot or dorsum of hand, there can be a little swelling associated with fracture.
So you should go for a x-ray of right hand to rule out any bony injury.
If it is any tendon injury then crap bandage should be used with painkiller medications for few days and if any fracture is there then POP slab should be applied for immobilization to reunite the bone, as per guidance of your orthopedic surgeon.
Dr. Naresh Jain