Going fullscreen without stretching in an XNA game

I've got a 2D game that I'm working on that is in 4:3 aspect ratio. When I switch it to fullscreen mode on my widescreen monitor it stretches. I tried using two viewports to give a black background to where the game shouldn't stretch to, but that left the game in the same size as before. I couldn't get it to fill the viewport that was supposed to hold the whole game.

How can I get it to go fullscreen without stretching and without me needing to modify every position and draw statement in the game? The code I'm using for the viewports is below.

            // set the viewport to the whole screen
            GraphicsDevice.Viewport = new Viewport
            {
                X = 0,
                Y = 0,
                Width = GraphicsDevice.PresentationParameters.BackBufferWidth,
                Height = GraphicsDevice.PresentationParameters.BackBufferHeight,
                MinDepth = 0,
                MaxDepth = 1
            };

            // clear whole screen to black
            GraphicsDevice.Clear(Color.Black);

            // figure out the largest area that fits in this resolution at the desired aspect ratio
            int width = GraphicsDevice.PresentationParameters.BackBufferWidth;
            int height = (int)(width / targetAspectRatio + .5f);
            if (height > GraphicsDevice.PresentationParameters.BackBufferHeight)
            {
                height = GraphicsDevice.PresentationParameters.BackBufferHeight;
                width = (int)(height * targetAspectRatio + .5f);
            }
            //Console.WriteLine("Back:  Width: {0}, Height: {0}", GraphicsDevice.PresentationParameters.BackBufferWidth, GraphicsDevice.PresentationParameters.BackBufferHeight);
            //Console.WriteLine("Front: Width: {0}, Height: {1}", width, height);

            // set up the new viewport centered in the backbuffer
            GraphicsDevice.Viewport = new Viewport
            {
                X = GraphicsDevice.PresentationParameters.BackBufferWidth / 2 - width / 2,
                Y = GraphicsDevice.PresentationParameters.BackBufferHeight / 2 - height / 2,
                Width = width,
                Height = height,
                MinDepth = 0,
                MaxDepth = 1
            };

            GraphicsDevice.Clear(Color.CornflowerBlue);

The image below shows what the screen looks like. The black on the sides is what I want (and is from the first viewport) and the second viewport is the game and the cornflower blue area. What I want is to get the game to scale to fill the cornflower blue area.