Does "I am Azeri" works too?
Absolutely and we accept it! :)
nein azeri geht nicht
I failed on my English!
True for me.
Are they mutually intelligible?
what about azerili, does it work?
This statement is not true. You can say 'Azeri' or 'Azerbaycanlı'.
'I am Azeri,' worked for me
I think that " I'm Azerbaijanian" is also correct