Why is my toilet not flushing completely?
What you might try is getting a new flapper since it is closing too soon anyway, also get the filling mechanism/float which will rush water into the tank quicker..they sell both items in a kit..and make sure the water level is high enough in the tank. You can even adjust the float a little higher. That will give the water more force from the added weight.   It sounds like all you need is a little extra push to get a complete flush.