Can I lay paper grocery sacks down under straw to mulch the vegetable garden for weeds and to help hold the moisture in?
You can do this, and the paper will eventually break down and get incorporated into the soil. It should suppress weed growth for at least two months, how much longer than that depends on rainfall. I think the reason that black plastic is used for a weed barrier is because it lasts longer.
Certainly. You could just fold them and lay them down or you could cut them apart and then lay them down to cover more of the area. They will keep any plant they cover from getting access to the sunlight all plants need to grow just like newspaper and cardboard will. How much moisture they will help hold in the soil will depend more on the thickness of what you use to hold them in place. 3 inches of straw will help hold more moisture then would 1 inch.