Milfshake:
We should be able to select a NEW ball... Or is that against some golf rule?
It's the One Ball Rule dictating no new selection during a game - a local rule which is applied for all PGA Tour Tournaments.
The RoG themselves only demand to play from tee to hole with one ball, except for admitted changes, e.g. when playing OOB or into the water. Between holes, players may select whatever they like.
An inofficial game not allowing the inter-hole exceptions is "Match Play Madness".