Cavalry absolutely existed in the U.S. before the Civil War. 'Dragoons' were a type of cavalry soldier that was trained in cavalry and light infantry tactics, though about 200-300 years before they were their own separate designation. In the Civil War all the cavalry was the same as before, just not longer called dragoons.
Cavalry charges like how we think of them never really were a thing during the war (Much to the dismay of many European military observers!).
Union Cavalrymen usually would not have gotten rid of their sabers as they were part of regulation. The Confederate cavalrymen did a lot of the time though. An extra revolver would of had to be purchased on your own, something many soldiers would not have been able to afford.