Are you sure you're not confusing AND and OR? https://en.wikipedia.org/wiki/Intuitionistic_logic#Relation_...

Literally copying from Wikipedia: The system of classical logic is obtained by adding any one of the following axioms: ϕ ∨ ¬ϕ (Law of the excluded middle), ...

where ∨ means OR, not AND... right?

Yes. In classical logic, p OR !p is always true. You have to add these axioms to get classical logic. Intuitionistic logic doesn't have that. p OR !p is not necessarily true in intuitionistic logic.

> A and not A are not necessarily contradictions

Nobody was saying p OR !p is necessarily true. The argument was about p AND !p. It really seems to me like you're confusing AND and OR...

The intuitionist will accept that "A and not A" cannot be true https://en.wikipedia.org/wiki/Intuitionism#Truth_and_proof