Good that you had your soil tested. As to double or even 1.5 digging, that will help to amend your soil. I've found that adding organic matter in the form of compost is one of the best ways of improving the tilth and drainage of clay soil. I don't know what Clay Soft or etc is, but the compost should be most helpful. I do not recommend peat moss. You don't say if you are putting in a veggie or flower garden, but you should find these sites helpful as well.
When weeding, the best way to make sure you are removing a weed and not a valuable plant is to pull on it. If it comes out of the ground easily, it is a valuable plant.