Self-employment and a partnership are two completely different things. What exactly is the organizational structure of your business?
Self-employment is included in your personal tax return, for both federal and state. You do not file a separate tax return for the business. A personal tax return, including self-employment, is federal Form 1040 and New York Form IT-201.
A partnership is a separate legal entity. The partnership has to file its own partnership tax return, separate from your personal tax return, for both federal and state. Information from the partnership tax return has to be entered in your personal tax return. A partnership tax return is federal Form 1065 and New York Form IT-204.
You cannot file partnership tax returns, either federal or state, with TurboTax Online. You have to use TurboTax Business (not Home & Business) for the partnership. TurboTax Online is only for personal tax returns.
I suggest that you sit down with a local tax professional to clarify what type of business structure you have, and what tax returns you have to file.