Commuting in wool and cotton is no big deal.
some of us ride to work in what we wear for the day. I ride minimum 6 miles each way, and wear my work clothes. And I put the hammer down, and arrive a little sweaty. and you know what? sweat dries out.
If you eat right you don't smell awful.
MAYBE you're wearing too much stuff? I rode this morning - light rain, 1 to 2C, in a wool T-shirt, short sleeve polo, a thin athletic jacket. Cotton shants, wool socks, leather gloves, thin cap. Leave the house underdressed is possibly helpful suggestion.