first thing i would do is figure out how to get those drains snaked because i'd bet that's half your problem. i'd even guess your custodial staff has one in their supplies as i've always know them to have at least one to test out before calling up a plumber and pay money. if not, you could rent one or better yet buy a cheap $10 25ft hand-crank snake from Harbor Freight (maybe even use a 20% off coupon since you can usually find a dozen in one Sunday paper) to keep in your classroom since this may be needed in the future.
if your sink doesn't have one yet, look into either buying or fabricating a sink drain trap to catch any materials that make it past an in-sink pre-trap. If you do some Google searching you can find many simple plans for making a sink trap out of buckets or rubbermaid tubs, etc and most likely under $20 in parts that will save a lot of future headache, or just buy one. for in-sink contraptions, many people have different setups for this type of trap. some variations i've seen are the "overflow tube" setup (like being described above with a tube that raises the drain and only lets cleaner water past it), busser tubs in the sink, multi-bucket setups, single bucket setups, etc etc.
one of the simplest ones I've seen is simply a 5-gal bucket with many small holes (1/8"?) drilled around the top lip about 1" down from the top edge. This 5-gal bucket sits inside of a busser tub, that was sitting inside the sink. 5-gal bucket is mainly for pre-rinsing hands and tools, when water level gets to the holes it spills over and flows into the 2nd tier/busser tub. The 5-gal bucket is deep enough to settle most of the clay and the 1/8" holes act as a filter; the 2nd tier is a secondary settling trap that helps keep clean water only going down the drain.
even just having a separate bucket or trash can full of water for doing a pre-wash on hands and tools could be the solution to your problem. ultimately though, teaching your students good studio habits will help the most. having a backed up sink is a good time for them to learn since they can physically see the repercussions of putting clay down the drain.