In order for you to apply for an inland spousal sponsorship you must first either be married or common law.
You can not drive up here with your car full of stuff, and your cat, expect them to allow you entry as a visitor, get married and then apply. You would need to likely come here 'light' (meaning store the majority of your stuff) bring the cat, a couple suitcases (something like a visitor would bring)and then seek entry.
Once in Canada either wait until you are common law (no working, no schooling, no nothing really) while you wait the full year and 1 day. You can THEN fill out your application for sponsorship and you will qualify for OWP inside of 4 months, full PR arriving roughly 12 months from application date.
OR
Once you are in Canada you can marry your girl, then you may apply for inland spousal sponsorship immediately. (or as soon as you can put a full application together) You will then be able to work roughly 4 months after your application is sent in.
If you do not wish to wait that long to work, you can opt instead to fly in for a visit, get married stay for a couple weeks then return to the US. You both can then get the application together and file an outland sponsorship. You can continue to live/work in the US and save for the move. You will be approved (barring any legal reason I am not aware of) in roughly 12 months (usually faster for US) and then you can pack up your car, your things and your cat and 'move' to Canada.