Yes. It is common to still be able to move a finger if it is broken.
Pain, swelling, loss. Of motion, deviation, rotation, stiffness, throbbing, numbness, discoloration, tightness all can be symptoms of a finger fracture. If one suspects a fracture, then elevation, ice , splinting and urgent evaluation is needed as many seemingly simple fractures in the hand require a detailed understanding of what to do to optimize healing and return of function.
Yes. A common misconception is that you won't be able to move a broken finger. That is not always the case. If you are at all concerned, see a hand specialist as soon as possible. Good luck.