If it is broken in the middle, then side cutters to the rescue, but how can we know where it is broken until we check, and after side cutters, it is too late. So i recommend the following.
The weakest point on the plastic gear tube is the connection to the smallest telescopic section of the antenna. This is usually were it breaks, due to flexing in the shape of a repeated "S" inside the tube (mental picture required), when pressing the antenna up against undue stress of residue, dirt, etc. If this is the break point, then the fix is easy. If it is elsewhere, then things are more complicated, but all is not lost.
Anyway, back to that "S" shape, which is formed inside the antenna by the plastic tube that should stay straight. The calapsing of the "S" is what binds inside the telescoping antenna, causing it to stay up. The motor makes it's noise, but either the slip function in the gear drive is weaker than the antenna's resistance due to it's gunk and residue build up plus the "S" inside it, or the drive tube is no longer hanging low enough to contact the round gear that draws it down....again, due to the "S". So the antenna stays up.
There are two screws as you mention which can be removed from the antennas base unit (they are not on the backside, they are on the front), allowing the coverplate to be removed, which usually covers the swirling up chamber (the plastic gear drives "layer" as it may be called since it is the villian here, and all villians need layers). Once this is removed, the plastic dish/disk/outter shell of the layer, can be easily removed. Now one can with patience and jiggleing press the antenna down (this is best done when activating the down funtion of the antenna by turning the stereo on, listening to the motor try to push the antenna up, with no result, then as it trys to pull it down, the additional downforce of pressing the antenna down, in my experience, has enganged the villian tube into its round gear drive, spitting the uncoiled plastic gear tube out into the trunk). Pressing down onteh antenna the whole time. Do not worry, it will coil itself back up later.
Refer to previous post for repair steps from here on out.
Turns out that many cars have the same unit design. I did this repair to my old 300ZXTT, when the plastic gear drive was broken about 12-14 inches below the top. I simply discarded the extra, reattached the new top of the plastic gear drive (broken end), then reassembled. The antenna then went up about 8 inches less, but worked fine. Why? Because there was an extra 6 or so inches in the layer at stock heights full raise, that was now being used. So, it was pushing the whole plastic drive gear up, through, and out the other side of the round gear drives teeth. BUT, gravity made sure it was there, sitting on those same teeth, ready to pull down, whenever they spun the other way.
Important point, when at my vacation house in Sweden where my car lives, I often have lots of free time between the rest and relaxation. So I take the time to do unessasary things like this, because I enjoy the challenge of repairing the unrepairable.
For me it is not a cost issue, or I would just buy another one. I do feel though that in the last 7 to 8 years I have met many NSX owners like myself that enjoy the challenge, increased technical understanding, and satifaction of a the completed job....and hey, if it saves a few bucks, then so be it. To each his own, and good luck too all you with this problem.