Hmm...
"I saw him alone"
That one is grammatical. That's a depictive, and something about the semantics of "alone" fits while the other adjectives don't.
So it's possible that there is some (lexical) aspectual restriction on the verbs/adjectives that can appear in the construction, so maybe it's not a good comparison with the verbs that more easily fit there?
"I saw him exhausted" -- this might work, if it is already established that at some particular point he was exhausted. (But that's a verb maybe?)
"I saw him sad" -- very hard to think of a context for this. Maybe some intense moment of sadness in contrast to other times in his life.
"I saw him blue" -- maybe like "sad" in the emotional sense, but for the color, hard to imagine-- I guess this does work if someone is in a temporary blue state-- "He wasn't getting enough oxygen and then I saw him blue-- it was terrifying" -- that
almost works (or some similar case of having temporary blue paint on one's skin), but it still seems awkward.
So maybe it's possible to have an adjective there. But for a verb:
"I saw him singing" it's just
so natural. Maybe it's because it's specifically progressive/imperfect?
It feels categorically different to me than the adjective cases though.
"Alone" works as well as the -ing forms, but it doesn't feel like it has the same sense to me in terms of predication.
Edit:
Adding "once" helps a lot--
"I saw him exhausted once."
And you can use other parts of speech too, at least prepositions:
"I saw him in a movie once." (Where he is in the movie and I'm not.)
or
"I saw him in a state of depression."
But are these the same as what's going on with the verbs? I'm really not sure.