When I check M1 or M14 adjustable rear sights and no matter if they are NM or standard, there are a few tools I grab real quick.
The first is a pair of what Brownell's calls "Magazine Tube/Cap Pliers." They jaws are tough but soft so there is no damage when grabbing hold of the elevation pinion drum so you can tighten the screw. I've heard these same pliers are used in the automotive trade and if so, they would be cheaper to buy at an auto parts store. Maybe someone from that trade will recognize these pliers and let us know what they are called in the automobile trade.
The next things I grab are two Brownell's Magna tip bits. I found these two sizes fit virtually every G.I. pinion and windage knob:
Bit #360-6, SD=.360, BT=.050 $2.92
Bit #240-7, SD=.240, BT=.050 * $2.92
You can find them listed on this web page:
http://www.brownells.com/aspx/ns/store/ProductDetail.aspx?p=417&title=MAGNA-TIP SUPER SETS?
You also have to have one bit driver handle and you could get that from Brownell's but they are also available at most hardware stores.
OK, so you grab the elevation pinion drum with the Magazine Tube Pliers and use the larger 360-6 bit to tighten the elevation pinion screw. Once tightened, you are done with that bit.
Then check to see if you have good audible clicks as you turn the elevation pinion up and down. If not, the pinion may not be tight enough. Then you use the smaller 240-7 bit to tighten the nut in the windage knob until it just "jumps" into place the first time. I've found this is almost the ideal position for most rear sights. Occasionally you have to tighten the windage nut one more "jump" if the sight is not tight enough with a elevation pinion that has a worn internal spring.
Now if you have good audible and repeatable clicks of elevation, try turning the windage knob. If you can turn it without wearing out your fingers that's good. Turn it right 8 clicks and back left 8 clicks to ensure the rear sight base moves according to the scale on the receiver. Then turn it left 8 and clicks and back to the right 8 clicks to see it does the same on both sides of the windage scale on the receiver. If that works properly, you are done with this step. If it is so tight you can't turn the windage knob, then you either turned the WK nut too far or the elevation pinion may be worn or you may need a RS cover (spring) that is tighter. You may also have to do some work on the bottom front of the aperture slide.
I think I should write an additional post on what to do next if the RS doesn't work, but that is going to be a long post and will have to write it later.