That only happens if you overcook it, and yes, I've been there, with a SC full of mushy grey meat and disintegrating bones, bleeeeeuuurgh.
It's fine if you watch the timings, I find 5 hours on high is about right.
I take it out, let the surface cool enough to remove the skin, I keep this to deep fry another time.
Then use the breasts to make dinner, always use the breasts first as they can dry out when being reheated.
Once dinner is finished it'll be cool enough to strip the meat from.
I use foil tubs and write roughly on them how much and what it's for, eg chicken thigh, pasta for two, small chicken scraps, risotto.
Then you chuck the bones back in with a carrot, an onion, peppercorns and a bay leaf, a glass of water and put on high for four hours.
Cool, strain, freeze.