Quick Fixes for Tenon corks.

The cork covering the tenons of woodwinds is usually natural sheet cork. This not something you are likely to find at your local hardware store. It is very fine and close grained. Over time it will dry and compress. This can make the joints wobble and likely make them leak as well.

If you leave you instrument assembled all the time this will happen faster. If it is dry where you live and you do not use cork grease this will also make them need replacing sooner. regardless it should be part of regular maintenance
 
I do my tenon and mouthpiece corks once a year. I always notice an improvement in the clarinet after doing this. Your instrument is just like any machine it needs regular service in order to perform well.Care of instrument tenons or joints is an overlooked item, it applies equally to flute head and foot joints and saxophones necks, all which must be perfectly, round where they connect. We use a series of pin gages to check this, but I digress.

If you use cork grease the corks will generally last longer as it slows their drying out. Though some greases seem to actually make the cork disintegrate. The grease also fills in the any small imperfections in the cork and compensates somewhat for minor out of roundness of the joint.
{mostip} No crazy glue it will make a horrible mess{/mostip}
If your cork is just loose there are a few things you can try;

first put a lot of cork grease on the joint and then heat it. The safest way to do this is heat a flat bit of metal that is about the same width as the cork or smaller and put it on the cork to melt the grease into the cork and expand it. Obviously if you heat it too much you will burn the cork.
 
I don’t recommend holding the instrument it over a flame either, as it is too easy to damage it.

So to reiterate heat the metal strip and use it to melt the grease. You don’t have to heat it very much

. If the cork is really loose or falling off, your next best bet is waxed dental floss. Wrap it around the tenon like you are wrapping a fishing fly. The wax will compress and you will have fix that will last for a while.

What about replacing it yourself?

That’s a possibility, though usually when I explain how to do this people realize it’s a lot easier and cheaper to take it to the repair shop. As I said earlier the first you thing you need is the cork. The cork is quite pricey at least $4-5 us for a 6x4 piece

There are lots of how to articles on the web here are 2 of them.
Links: How to By Steve Fowler for clarinet http://www.fowlermusic.com/clarcork.html http://www.fowlermusic.com/sax_neck.html and for saxophone