Can I add someone to my deed without the house being fully paid for? If I can will they be responsible for the mortgage.

You can... but its stupid.. they will OWN a share of the house you can't take away, that they won't even have to pay for. And it has NOTHING to do with who is responsible for the mortgage. The people responsible for the mortgage are those who have their names on the mortgage.

Normally you may have to get the lenders permission before you can add them. And no, if you just add them to the deed but not the mortgage they have no obligation to pay for it.