I hardly ever sweat and my skin is so dry when they say shoe leather well that is me and I am a woman about 60. I have had this most of my life my father was just like this and my 3 brothers. I have a thyroid disease I take Levothyroxin and have for many years. I eat a lot of snow cone type ice the cup I use is 64 oz and I eat this at least 7 to 10 times daily it is a craving for me. I use non perfumed and sometimes patrolium Jelly which is so greasy and does not work well anyway it seems nothing does. I have diabetes, Celiacs disease, diverticulitis, varicose veins in my throat, high blood pressure among a few others. I need to find something that will work.
In my experience, petroleum jelly is pretty much useless for the purposes of alleviating dry skin. I'm lucky to not have dry skin but people I know who do, suggest coconut oil.
As NGE suggests... anti-inflammatory / autoimmune. I would suggest to start: no sugar (even hidden sugar), no bread, no pasta, no wheat, no grains, no dairy, no eggs, no legumes, limited fruit, no fruit juice.
I've kinda muddled Paleo & some other concepts but I have my reasons. My wife has had problem with improperly diagnosed autoimmune issues for a LONG time culminating in a diagnosis of sarcoidosis. She went the traditional RA / steroid approach for a while....no real improvement.
She switched to a non-traditional MD who recommended the above, WAY before I stumbled onto the slow carb / no refined carbs concepts to address incipient blood sugar issues, which eventually lead me to Paleo.
Anyway.... this VERY restrictive diet calmed her autoimmune issues & allowed her to function again. She's has reintroduced eggs, very limited dairy & limited fruit. Otherwise she is pretty much Paleo overall. It keeps her "flares" at bay and we figure "if it's working & easily doable...don't mess with it". It might not be "Paleo pure" but it works.
She ate a LOT of wild salmon & green leafy veggies, broc, sweet potato & cauliflower but not much else. A lot of dinner for breakfast. Yoga has helped as well. She lost ~30lbs of fat.
I think you find Paleo quite doable & enjoyable.
Coconut oil is a wonder solution for dry skin. Rub it in your skin before sleeping for a few nights and see the change. You can even try any parachute product as they all contain coconut oil/milk extracts. As it is summer, I would say go for their parachute summer fresh body lotion. It works well for dry skin without causing any irritation.