Did the U.S. Cavalry have a swordsmanship training program?

I know they did have drills that they ran from horseback.

For the most part, Cavalry did not fight from their horses. They did when doing a charge, but other then that, they would dismount if they had to fight. They were not intended to fight, cavalry was intended to find the enemy and guard the flanks of the moving army. Doing raids was a secondary job of theirs.

If they were fighting with sabers on foot, they were in trouble. When they did use their swords, it was in a charge and from horseback. So the drills were based from horseback. That is why cavalry had sabers instead of swords. Sabers were curved so they didn't break or get stuck in the target on impact from a horse.

Infantry and Artillery Officers used swords and would take some fencing lessons.

Enlisted probably focused more on their carbines and pistols then their swords.

I know the Army had a qualification badge for it just like we do for other weapons today.