I broke my ulna and radius. Will this affect my ability to use my hand?

It certainly may. The break (fracture) extends into a joint itself in an adult. In such cases it may be very important to restore the joint by surgery. In children it is not rare that both forearm bones are broken like a green stick snapping (we call these "greenstick" fractures); usually treated well in a cast, often including the elbow. In adults, "both bone" forearm fractures usually require surgery to fix.
Radius/ulna fracture. If proper care is instituted and you heal uneventfully, it is unlikely that you will sustain any significant impairment.