Howdy
Your question addresses the reason why in CAS once a hammer is cocked, the round must be fired, it is against the rules to lower the hammer on a live round.
Anyway, I suspect you are loading all six chambers. Instead, try just loading five.
With a traditional Single Action lockwork like your USFAs, the best procedure is to set the hammer at half cock, open the loading gate, load one, skip one chamber, load four more, close the loading gate, cock the hammer all the way, and then lower it onto an empty chamber. If you do it correctly, when you lower the hammer all the way it will be lowered onto an empty chamber. This is also the way to avoid causing a ring around the cylinder. By pulling the hammer all the way back and then lowering it, you are resetting the bolt. The click you hear when you lower the hammer is the bolt resetting. A bolt leg is popping over the hammer cam and resetting itself.
Loading a traditional single action with all six chambers is not a safe practice because if you happen to drop the gun and it lands on the hammer, either the sear or the 'safety cock' notch on the hammer is likely to shear, and the gun will probably fire. Loading the way I described you can peer at the gun from the side and see that there is not a cartridge under the hammer, there will be a space where there is no cartridge.
I suggest you get yourself a set of snap caps and practice this at home rather than trying it first time at the range.
Now, the time is going to come someday when you need to lower the hammer on a live round. Or, you should just get practice lowering the hammer with your snap caps. I always lower the hammer on the empty chamber, I never pull the trigger.
I suspect that your USFAs have the same high hammer spur as a Colt. Here is a sequence of photos showing lowering the hammer on a Colt. Note, I am using my left hand because I needed my right hand free to shoot the photos. It would be the same with my right hand.
Basically you use the hollow in the hammer spur and roll your thumb though it to control the hammer on the way down.
In this photo, the hammer is at full cock. Notice the position of my thumb, rolled forward a bit into the hollow. Notice too that my finger is off the trigger.
In this next photo I have forced the hammer all the way back against its spring, so I can control it when I pull the trigger. I have started pulling the trigger, but it has not released the hammer yet.
The trigger is fully pulled back and I am easing the hammer forward, controlling it with pressure from my thumb.
I continue to allow the hammer forward, rolling my thumb back in the hollow.
The hammer is almost all the way down and I am easing the pressure with my thumb to allow it to get there.
The hammer is all the way down. My thumb is no longer bearing down on the hammer and I am lifting it away.
Buy your self some snap caps and practice doing this. You can also practice it with an unloaded gun. Don't try it with live ammo until you are confident you have mastered it. It really is not hard, just takes a little bit of practice. Next time you are watching an old Western, watch when an actor lowers the hammer on his gun. If an actor can do it, anybody can.
P.S. If you have fired a string of shots but the gun still has a few rounds in it, and you want to change targets, I would never lower the hammer and walk in front of the gun. Never. Always empty the gun before you set it down and walk in front of it. Either shoot it empty, or set it at half cock and unload it before walking up to change targets. Set it down and leave the hammer at half cock before changing targets.