I get that way year round.
If you use an antibacterial pump soap or purell-type product frequently during the day, STOP. They contain ingredients which hasten drying of the hands, and really shouldn’t be used by folks with sensitive skin. Switch to a natural bar soap that contains glycerin, or use something like Dr. Bronner’s for a pump soap (also veg glycerin) - it’s extremely gentle.
Be picky about your lotions. Lingering slickness not only makes your hands feel gross when knitting, it may be a sign your lotion contains ingredients that are actually drying to the skin (they do this deliberately, so that you’ll buy more). Stick with lotions that are thick, yet absorb cleanly. Eucerin is a good one, esp. if you have sensitive skin. Avoid things with heavy perfumes, because they will contain excess alcohol which = drying. 3M carries a brand called Cavillon - fragrance free, hypoallergenic. It’s used alot by EMTs and health workers because it won’t cause latex gloves to disintegrate (CHG compatible). The only lotion that has worked better for me than Cavillon was one I made at home. Best thing: available in small bottles, so you can stash one in your knitting bag.
Softening rough hands. You want to exfoliate the snagging skin, but you don’t want to damage it more with ingredients it doesn’t need. Skip anything containing alphy-hydroxy.
My recipe: take 1/2 cup of granulated (table) sugar, 1/2 cup of vegetable oil (I like using jojoba, but your mileage may vary). Olive oil is a good choice, corn oil notsomuch. Mix it together, scoop it up with your fingers and rub it through your hands as though you were washing them with soap. Do this for a minute or so, then rinse off under warm water (do NOT use soap). Gently pat dry, then seal the moisture into your skin with your hand creme.
That recipe can be used on feet, hands, face and body. And it works, trust me. I wouldn’t be able to survive winter flareups of psoriasis without this regimen! It works better than Gardner’s Hand Therapy by Crabtree & Evelyn, and is tons cheaper.