What can I do to stop my colored pencils from falling apart?
Prismacolor pencils are notorious for lead breakage, but the creamy lay down keeps folks coming back. However, there are some things you can do to moderate the problem, in general (I'm going to answer a bit beyond the question).

Pencil Selection

The pencils that tend to break more easily have off-center cores. If you're buying open stock, take a look at the bottom of the pencil and ensure that the core of it is in the center of the pencil body and not askew. None of mine are in this state, or I'd post a picture, but it's really obvious when you see it.

Sharpening Technique

A little counter-intuitive, but I've found that instead of rotating the pencil in the sharpener, going the other way works better. In this scenario, hold your pencil in your "off" hand and then using your normal hand, rotate a very sharp sharpener around. This should be very sharp, the wood should shave off in a long, continuous strip. If it doesn't then it's time to replace.

In addition, don't create a long point. Smaller amounts of exposed core tend to be less fragile and so don't break as often. Find a sharpener that lends itself to that or allows you to control the point.

Fix It Up

Prismas are wax based and can sometimes be fixed by heating the core to fuse it together again. Prismacolor doesn't recommend the microwave, rather a sunny window spot, but many use the microwave. Just keep it short, seconds at a time. After all, you're not looking to make it runny, just softened enough to fuse again.

Handle With Care

Since the lead is fragile, avoiding drops and bangs is a must. The problem, of course, is that transportation to your supplier probably banged and bashed them around, causing the odd one to break even before you got it. To that end, I really recommend going with open stock since you can check the pencils there and, as I noted above, you can confirm that it's not askew.

Change 'em Up

At the end of the day, the price of Prisma feel is the weakness of the lead. If you're reaching the point that it's too much, then there are alternatives. Faber-Castell and Caran d'Ache make great pencils and they don't have the breakage issue. That's partly because they're vegetable oil based rather than wax, but also the lead is bonded to the wood, so even if there is a break inside, all is not lost.

In any event, I have no familiarity with Caran d'Ache pencils to use, but the Faber-Castell Polychromos are really very good and also available in open stock. For me, open stock is a must. So, I have the full Polychromos set and a select bunch of Prismacolors (I generally prefer the white Prisma). They work together just fine.