You need to understand that the medical process for enlistment is different than for officer. Recruiters typically work the enlistment side. OCS is the exception. ROTC and SA candidates go through a different process (BGO for USNA), not recruiters.
Your Mom will not work because they will require you to fill out your medical history and if they give you a remedial or DQ, they will want your family physician that is on record. Plus they may send you to their DoDMERB specialist.
Eczema and psoriasis are 2 different things. Please read this link.
http://healthguides.healthgrades.com/article/psoriasis-eczema-dermatitis-whats-the-difference
Psoriasis is a lifelong condition that causes thickened, red patches of skin and silvery-white scales. Psoriatic patches can be itchy, sore, and even burn.
Eczema usually shows up as very itchy patches that become red, swollen, and cracked from scratching. Rashes usually occur on the face, inside the elbows, behind the knees, and on the hands and feet.
Certain substances may trigger an eczema breakout, so it’s important to identify and avoid them. Common irritants include household cleansers, detergents, soaps, chlorine, and wool. Stress and abrupt temperature changes can also cause eczema to appear.
To confirm whether you have eczema, your doctor may ask about your family history of allergies and want you to have an allergy or blood test.
Hence, why you should go to a dermatologist to determine if it is one or the other or neither, and it appears you are self diagnosing.
I get you want to be a fighter pilot. However, here is the thing, you can get Qualified for USNA, but DQd for rated. You will need to pass the FAA Flight Class 1 physical which is a higher level than the DoDMERB. You need to also ask will you be happy in the Navy if you don't fly.
You are 13. Get an appointment and commission, but no rated slot, you will still be in until at least 2029. It will be 9 years of your life (4 USNA and 5 Active Duty).
I am not trying to dissuade you at all. I hope you go onto fly the 35, but you need to sit down with the folks now and tell them your desires. You need their guidance as parents, not only from a medical aspect, but from the competitive admission aspect. Mom being a doctor knows and understands that long run academic process. To become a pilot it is 2 more years of work/academics before you are operational...on a good day.
~ She will be able to talk to the dermatologist one on one from a peer aspect.
It is great you are getting in front of this, but to really be in front of it, talk to the folks and get a true diagnosis by a dermatologist.