Is it my writing partner and myself or my writing partner and me?

The correct phrase is "my writing partner and me." You use "me" as the object of a preposition (like "with" or "to") or when it follows the verb in a sentence. To check, you can remove "my writing partner" from the sentence; if it still makes sense with "me," then it's correct. For example, "He gave the book to me" sounds right, while "He gave the book to I" does not.