Keeping the main thing, the main thing.
The purpose of a school library is to help students and teachers find the resources they need. I get why genre selection is popular. It lets kids go to one section, which they know already, and find a book most likely to pique their curiosity. Yet it also prevents them from stumbling upon a book that is outside of that particular genre. In addition how do you shelve a Katie Kazoo book? Is it humor or realistic fiction? What about Gregor the Overlander? Is it fantasy fiction or adventure? Let's not even talk about the Hunger Games. It could be fantasy, science-fiction or adventure?
So if the main thing is to help kids find what they are looking for we need to examine a few things like equipment, time allotted, and methods used.
Equipment
Most schools by now have converted to computer catalogs. Ours uses Destiny Online. Are you students allowed to use them each and every time they come to look for resources? We have ours is set up and billed as a one stop shop for materials and websites thanks to Webpath Express. That said we add additional useful websites into our collection in addition to the ones we pay for. Do you have visuals of the book covers? That tends to be a great help. Again, this is likely an option to pay for, but well worth the money.
Time
Do your students have time to browse? If your answer is yes, then are they using their time efficiently? Do they spend almost all of their 15 minutes on the computer and one minute at the shelves? Do they talk to their friends for the majority of the time? If you answered no, what can you do to convince teachers to allow their students more than 5 minutes in the library? If your check out is during class times, how can you adjust your lessons to accommodate this and still accomplish what you need to do? Perhaps you could have students check out in small groups during class to discourage talking and rotate through the class so not everyone checks out at the same time?
Instructional Methods
I saved the methods used for last because it involves taking a hard look at ourselves and the manner in which we instruct. This can be difficult at first but one thing I learned a long time ago is that reflective teaching means better teaching. According researchers like Eric Jensen a lot of kids are simply bored in their classrooms and thus aren't paying a whole lot of attention when we teach. This is true of all instructors not just librarians. So when we teach are we talking the majority of the time or are the kids talking about the tack in front of them? Are they actively engaged or passively waiting for you to tell them what to do? How often do you go over the process? Do you embed it in multiple activities throughout the year so that you aren't necessarily having a DDC lesson? I tend to embed the OPAC in most of my lessons so that finding resources of any type become second nature for a large number of my students.
Conclusion
For the record, I don't recommend having your students memorize Dewey. I don't know everything and I'm in the library every day. It's a worthless skill. The reason we used to do that is to find the material. Now, we type it into the computer and instead it tells us where to go. As long as your students understand which section is which in your library then they should be able to navigate through the shelves to the right place.
I know there are still some kids who like to browse. That's great. How well are your shelves labelled beyond the numbers? Do you have visual aids to assist them? One librarian has some live cockroaches above her insect section. I don't do well with live critters but I think the message is awesome. I'm working on signage for my shelves. Each column gets a sign with graphics of what is on the shelf. Give them visual clues, not bookshelves that look identical from a distance.
I've probably not convinced anyone to change their mind, but realize that first and foremost changing over your library organization method is a lot of hard work. That in itself is not a reason to refrain from doing it. It also isn't something that vendors tend to catalog for so there will be continuous cataloging for your library personnel, whether that means you or an assistant. It will also mean lots of money spent on genre stickers. If you are seriously invested in the idea and you are convinced that it will increase your students' ability to read then get to it. I support your bold effort. I'm simply going to work to make students and teachers better able to navigate the system we currently have.
