From the pictures it looks like the white shelf sits a dado (groove) in the rails. Unless you can separate the rails from the uprights, I don't see you getting it apart.

Other than one picture I don't see where you took out the screws. In pictures 5 and 7 there seems to be some kind of cover in the upright, maybe for screws? In some cases this kind of stuff is assembled with a screw and dowel that's glued in (the screw holds things together while the glue sets up). You might be able to beat it apart but if the dowel breaks, you'll need to drill it out to insert a new one.

It might be just easier to mask off the wood and paint the shelves. The other option might be to cover the shelves with something like the stuff they use for cabinet shelves.

When you removed the screws did nyhting becaome loose? If not, then your answer is - the pieces were glued first and then screwed.

I'll second what mike said: it's probably easier to mask of the wood and paint the shelves or just cover the shelves with contact paper.