I can't relate to your entire situation being a man in a more industrial job where my only clothing requirement is to arrive at work in a company logo T-Shirt, but I'll try to help a little bit.
I would start by either leaving your work clothes at your workplace or rolled up in your backpack, panniers or whatever. Don't worry about what to wear to work when the roads are sloshy. I wear a pair of swimtrunks on the ride in with my wallet and phone double baggied inside my backpack..
As far as arriving to work sweaty and smelly just take a shower before your ride. Sweat doesn't smell on a clean body. Do a wipedown in the ladies room with Wet Wipes before changing. Don't try to break any speed records in the morning (save that until the ride home) to keep the sweat down. Obviously, as a women you're going to have different issues than a man, such as longer hair, makeup etc, so I can't relate to that.
FWIW, I wouldn't wear pants under your skirts. If it's sloshy you're still going to get your workwear dirty. Hope I've helped some.