You do NOT need a passport if you are driving over the border, and flying from Buffalo to a destination within the USA. You will need photo ID to get through security, and you will need a government issued photo ID PLUS your birth certificate at the border crossing. The above is stating that you need a passport only if you are flying TO or THROUGH the USA FROM Canada, or flying back INTO Canada FROM the USA. USA to USA destinations by plane only require photo ID.

I'm assuming you're not flying from Buffalo to another destination outside of the USA.
No worries, you're fine, as long as you have the required photo ID and birth certificate. However, I highly recommend getting your passport very soon. It makes life so much easier when crossing the border.