Is it possible to use the possessive s with countries like
new york's state
oklahoma's state
or just possible to use
the state of oklahoma????
You would write these as New York State, Oklahoma State, etc. The genitive "of" is also correct, i.e. The state of New York but "state" would not be capitalized when used together with "of".
In addition, New York and Oklahoma are states; they don't own states, which is how "New York's State" and "Oklahoma's State" are read. Apostrophe -s is not required. It changes the meaning.