What happens if the hammer drops on a spent case? Load up 1 or 2 in the cylinder with the rest of the rounds live. Better yet, have someone else load so you don’t know which chamber won’t go bang when the hammer drops. If the gun sits perfectly still when you get click instead of bang, then it may be the sights. If the front sight moves on the “click,” it’s most likely something you’re doing in anticipation of the recoil. Safer to do this experiment with a semi auto because you can have someone randomly load a couple of dummy rounds or snap caps into a full mag, and hand you the mag, instead of passing a loaded revolver around.
If you determine that it's the sights, you can lower your rear sight and/or raise the front sight (unless it's a dovetail, in which case you'd need a taller front sight) - Bullseye target shooters tend to train with their sights "level and centred" just below the bottom edge of the black aiming mark (image 1), with the POI in the 10. My personal preference was (is? I don't shoot much these days) "centre hold" (image 2), but the Walther GSP rear sights wouldn't lower enough to permit that, so I crushed a .22 case with the rim cut off, drilled a hole, and screwed the front sight down on top of that as a shim (trimmed to fit, of course).