Am I slow cooking chicken correctly?
Yes, you are slow-cooking correctly.
If you were cooking some chicken adobo then having the meat fall off the bone is intended. 

Otherwise if you want to serve if while everything is still attached, you can do so anytime after you have simmered or poached the chicken (making it safe to eat), not sure it qualifies as slow cooking however. As pointed out by @Aaronut, extended cooking will break down connective tissues - it breaks down collagen.


If your chicken is fat, then it may take longer to break down.
You can develop a caramel crust on your chicken so that it holds longer and lock in the juices.
tip: put the fattier part on top, so that it runs down the chicken.


Here is some information about searing; it can cause a maillard reaction which does a few interesting things:


it will add more complex and varied flavors, by reacting the sugars, amino acid, and heat.
caramelize stuff, e.g. onions :D


In some cases, pressure-cooking can achieve the same results as slow-cooking in less time, while making it easier for the maillard reaction to happen.