How could I tell if I have broken my thumb?

X-Ray is best way. Taking a simple x-ray is the quickest way to determine if there is a fracture. A walk in clinic equipped with an x-ray machine should be less expensive than a hospital er is you do not have insurance. If you have a physician, an exam could steer you to the right place, perhaps to an x-ray facility with a script in hand for the x-ray. Missed fractures may lead to complications like displacement.
Finger fracture. There is no way short of an xray to tell if a thumb that is injured has a fracture. That said an obvious deformity after trauma is a good indication that treatment is needed. Just saw a patient this week who neglected to get his thumb fracture treated . It healed in a malunion. Now he needs a reconstructive procedure. So if it hurts, doesn't move, and loks fuuny get it looked at.