Hey Wes,
I actually encountered this exact same problem recently. I picked up a 586 no dash at my local pawn store and when I brought it to the range, I had 3 clicks where I should've had booms. All 3 of the clicks were on the 6th round of the cylinder. My gun also had not had the modification done.
I called over to Smith and Wesson about sending the gun in for modification and mentioned to them about the light-strikes. First of all they said they are not accepted any more work until after August 12 of this year. Second (and most importantly), they said the light-strikes have nothing to do with the recall/modification.
The advice I was given from S&W was to remove the grips and tighten the mainspring screw on the inside of the grip frame. I have not taken the gun back to the range to test fire yet, but I can tell there is a lot more tension/stiffness in the action, and I believe this tension will provide enough power to give consistent primer strikes throughout the entire cylinder.
You may give this a shot first before spending money on repairs. If you pull the trigger and the hammer falls on a live round that does not fire, keep that round lined up with the barrel and hold it on target for 60 seconds just to insure that you do not get a delayed fire. It can be very dangerous to open the cylinder or to turn the cylinder that contains a hot round that still has the potential to fire.
I'll update once I've had an opportunity to test the screw tension at the range, but this may be a starting place for you before spending additional money on your new purchase.
Best of luck, and be safe!