I bring my clothes in before and I have clean up supplies at work. I ride my road bike, which only has a small seat bag, because of the geometry and size of my bike. I don't have a shower; I shower before I leave and then when I get to school I wash up (face, pits, anything else that feels sweaty) and wet my hair down, use gel. My hair is short, so it's easy. Then I do my make up. I use my Camelbak that has a backpack to bring papers, lunch, and other small things. I take the bladder out for more room.
You might want to try doing a partial commute. Drive somewhere that is about half way with your bike, park and then ride. I did this last year, after I moved. My old commute was 6.7 miles one way, so this is a change. You might want to also try riding different routes on the weekend, to test them out.
I checked out an alternate in the car this morning. It's at least 5 miles shorter and no climbing! I will ride this way in the AM and then take the route I did yesterday home. Hoping to ride in tomorrow, even though it's supposed to be 35 in the morning(but 70 in the afternoon).