Clever idea but I am not sure how well it would work in practice. The problem is that your adjustment is unidirectional i.e. your nuts can move apart automatically but cannot move back together.
Take this scenario with the initial condition that your screw is 10 tpi and two nuts are exactly one thread (.100) apart where they are threaded onto the screw. Now let's assume that the screw is a bit inconsistent so as you go down the screw the distance between the thread crests varies some. So there are sections where the threads are perfect, sections where they are slightly too far apart (.105) and sections where the threads are too close together (.095). This can be due to the way the screw is made, wear, or both. Well too close together can't be because of wear but you get what I mean. If this weren't the case, this entire discussion would be unnecessary.
Now as your nut goes down the screw, it hits a section where the threads are a bit too far apart and the gap between the nuts expands and the screw takes up the difference and locks in the new gap. So far so good. However at some point, you have to hit a section where the threads are closer together. But your nuts are set for the greatest gap and the nature of your adjusters is that they can separate the nuts but can't bring them closer together. Therefore your screw will jam.
This is why the spring works so well: it can dynamically adjust the gap between the nuts but at the cost of some wear as you have pointed out. As well there is almost nothing simpler and more reliable than a belleville washer.