I'm deadstoned and I'm going straight to bed.
I have never heard of this. If it exists as a slang expression, it has a very narrow usage, so I wouldn't recommend using it. Where did you learn this? Perhaps you were thinking of 'dead beat'?
I caught the plane by the skin of my teeth
Yes, this makes sense.
I don't like my current job but I suppose I'll have to make do with it.
I can see why you've said this, but it doesn't feel 100% natural to me - perhaps because the obvious phrasal verb to use here is 'put up with it'.
For some reason, we rarely say 'make do with it' - we either say 'make do' on its own, or 'make do with..' followed by the thing itself. 'Make do with' is used when we have to substitute one thing for another, and the substitute is OK, but not as good as the real thing. For example, if you have no wine to add to your sauce, you can 'make do' with some vinegar and sugar instead. It's not as good as the real thing, but it's a compromise.