"Besides" is inclusive, "except" is exclusive [duplicate]
Solution 1:
Look in the Merriam-Webster dictionary
besides (prep)
1: OTHER THAN, EXCEPT
Nothing besides a miracle could help them.
2 : TOGETHER WITH
a decision that, besides being practical, is morally right.
I believe you are correct that the first definition (other than, except) applies in negative contexts and the second in positive contexts.
Deciding whether something is a negative or a positive context can be tricky in English, so it might be possible to come up with a sentence that is ambiguous between the two meanings.