I actually make my own shampoo for my grooming business. It's made specifically for dry skin and dandruff; inspired by my older mare who has both. I call it "Mud Spa" and it's a soaker (you leave it on for 10-15 minutes and then rinse it off). It does wonders for the skin and coat and rinses off super clean despite the mud, haha, but it tends to fluff them a little too; it lifts the hair and gets underneath. So they aren't slickly shiny afterwards, but in a day or so can be. It's great for relieving itching too in my experience. However, the formula's a secret ;)
Not on here to sell stuff but I can post details to order if anyone wants some. However, I've found honestly the best way for a shiny coat and healthy skin is regular, thorough grooming! Between baths on my girls I brush in the following order: a curry comb (metal in the winter/ spring, plastic in the summer/ fall), hard bristle brush, medium bristle flick brush, then final go-over with a super-soft finishing brush (usually goat's hair or similar). This is also the best for shine; it brings the oils in their coat to the surface so they are really slick looking all over. You can use conditioning sprays too for shine and moisture; or add conditioner to their bath shampoo water for all over moisture. If they are super itchy I'd do this regularly, at least once a week. If the flakes are big or their skin is inflamed you may want to invest in a medicated shampoo for horses and do the same, leave it on for 10-30 minutes depending on severity of skin issues and patience of your horse before rinsing it off.