Susan is the roof really leaking? If you have not had this place for years is it possible that the roof leaked at one time, roof over was done and the ceiling painted without proper primers. And now the stains are bleeding through the paint. Not trying to say that your roof isn't leaking. But i have seen people mistake a stain bleeding through paint as a current leak.

Have not tried to use sandbags to hold down plastic sheeting on a roof but i don't see why it would not work. If i was going to try it i think that i would still put wood around the edges and put the sandbags down on the wood so that the whole length of the edge is held down which will make it harder for the wind to get under the edge of the plastic. Not positive that it will work but i would try it.