In this section, you'll add to the background by duplicating the existing text layer, animating the new layer, and changing it to a supplemental shadow. You can easily do this using a only a few additional effects:
Duplicate the Random Text Basic layer, and rename layer 2 Fill Text.
Change the Fill Text layer's Scale to 150, 150%.
Apply Effect > Blur & Sharpen > Box Blur, and set the Blur Dimensions to Vertical and Iterations to 3.
Set a keyframe for a Blur Radius of 0 at time 0;00.
Go to time 2;15, and change Blur Radius to 55.
Expand the Fill Text layer's Fractal Noise effect, and change the effect's Blending Mode to Hue.