I have two ways of resolving this for the types of bags that I use, both of which are superior to running with a backpack.
For the cloth SE Asian style bag, I just pull strap section forward and tie a knot. Depending on what I'm carrying, this is usually tighter than a backpack without restricting my breathing by pulling on my shoulders. This is by far the easiest way to run. I've made 4-5 mile runs on short notice like this.
The other, for when I have a bag with longer straps such as my leather messenger bag or my map case is what I call the Drill Instructor carry, which is to pass my arm through the strap and over my head. The loop effectively converts the shoulder bag into a backpack and depending on the direction I reloop, I can carry my load between my shoulders or on my lower back, depending on the amount of weight or what I'm carrying.