There are several things I'd take note of related to your setup and what you appear to be attempting.
First, the shoot through umbrellas are, by definition, diffused, uncontrolled light sources. The intent with using a shoot through umbrella is to get light everywhere. If you want to keep the light off the background you need to be using softboxes with grids. A large, gridded soft box can be aimed to minimize or eliminate spill onto the background. If spill is still a problem because of space limitations, then using a large V flat or similar gobo device can block the light from the background.
As for keeping the background in focus. You are running into the limitations of optics. Using a 5DII with a 70mm lens 10 feet from the subject according to DoF calculators such as the one I have on my iPHONE, F11 offers 4 feet depth of focus. At F6.3 the depth of focus is 2ft 3in.
The 70mm focal length is kind of short if you are planning only head and shoulders portraits. Unfortunately longer focal length means shallower depth of focus as same camera to subject distance. For example a 100mm lens at F6.3 at 10 feet has a DoF of 1ft, 1in. Which is just enough to keep the face in focus. Also, remember that the DoF is both in front and behind the point of focus. For the 100mm example its 6in in front and 7in behind. For the 70mm example its 12in in front and 15in behind. In both cases that rear distance is well less than the distance between the posing stool and the background.
Generally, when shooting portraits, its intended for the background to be out of focus. I don't remember seeing a post explaining why you've decided you want it in focus.