Is it correct to say I were?

Answer 1

I were is the subjunctive first person of 'be' - but is certainly still encountered.

If I were a carpenter

And you were a lady

I were will only ever be found in counterfactual statements (statements where what is being said is not currently true):

If I were Santa Claus ... (but I'm not)

I slept as though I were drugged ... (but I hadn't been)

Answer 2

No, "I were" is not grammatically correct. The correct form would be "I was" for singular subjects or "we were" for plural subjects.