I have a lot of experience dealing with minor acne, and have friends who dealt with major acne. Most of what you hear people say about acne is utter myth. In fact, many of these myths were taught in school systems until recently.
Dermatologist now have a fairly complete set of tools for handling any level of acne until you 'outgrow it'. In some severe cases, Accutane (Isoretinoin) may be required. This works on around 90% of patients. It is also being investigated for long term health problems. It shouldn't be used for minor acne in most cases.
Probably what you need is to go see a dermatologist who will likely A) Give a topical Anti-biotic B) Give a temperary oral Anti-biotic C) Give a pore unplugger D) Recommend appropriate soaps/moisturizers such as Purpose, Cidefel, Vanicream all make soaps/moisturizers with no dyes or fragrances.. note that these same companies also make unsuitable products. This will likely render your skin acne free and put you at zero risk for acne scaring which is painful and expensive to try fixing later. There is also some new light treatments which I cannot speak for since no one known to me has tried them.
Your toothpaste containing Sodium LaurylSulfate may also be causing/aggrivating acne and other skin problems. Rembrant and Biotene make toothpastes that are free of this, note that the toothpaste was still contain a flouride to be effective (both of these do).
Finally, you may need to use a detergent that is free of dyes and fragrances. I use Cheer Free. It is best to use a Powder.
These are long term solutions. They do work extremely well, but you have to actually follow the directions over a period of time. That requires faith that it will work. Most people do not follow the directions over a long enough period of time and then spend the rest of their lives trying to convince others that modern scientists are all "quacks". Right... and we all know life originated from an incestious relationship from Adam and Eve's offspring