Hi Jacob and welcome to the dreadie-life!
Anyhoo, here is a little reiteration with some redirection:
The Nutrogena Anti-Residue should only be used to remove gunky buildup, like a clarifying shampoo, it should not be used as a regular shampoo.
There are various alternatives that are great for keeping the scalp clean such as the BS/ACV (baking soda/acv otherwise known as "no poo") which you can find a recipe for in the dreaducation pages, Dr. Bronners, and the dreadlock shampoos from Bucks County Soap (these are the most popular but, by far, not the only available.) With the first 2 it's good to add essential oils for certain effects.
When you wash, only wash the scalp with the fingertips (no need to lather the entire head up or scrub at the hair itself) and the water will rinse the shampoo through your hair, in turn, cleaning it without disrupting the process. With the BS/ACV option, you don't have to do any rubbing because it's a "no-touch" way of cleaning.
With drying, gently squeeze excess water and/or flip the hair about while still in the shower. I even use a towel to squeeze and soak up the water since my hair holds a lot of excess but no rubbing.
The first sign of progress will be your hair starting to form it's own sections. Over time, these sections will start to get fuzzy and then knot up in places. Once this starts, it's your job to pull the sections apart so that they don't get out of control and start to dread together. The all-natural process could take months to years to dread, depending on your hair type.
As for the beanie, it's going to be too tight to allow your hair to move to form knots. IF you do want to keep your hair covered or out of your face, replace the beanie with a tam. Tams are essentially saggy beanies. They are big enough to allow the hair some movement when worn.