1) I work in an office that requires business casual (button down shirts, slacks, dress shoes, tie optional)
2) I commute daily by bike 7 miles in around 25 minutes
3) I live in Gainesville Florida, home of The Swamp, so I know hot and muggy
4) I do not have shower facilities at work (yet, but I'm working on it).
I leave my shoes & belt at work, so that's less to carry. I also pack my lunch considering weight more than what I want to eat
. Less weight=less work=less sweat. I starch my shirt and slacks and roll them up tight in my pannier. I generally arrive to work 30 minutes early, and just sit at my desk and cool off while replying to emails, etc. About 5 til, I'll make the walk to go change in the bathroom (which for some reason is not A/C at the same level as the rest of the building). Splash water on face and anywhere else that may need it, paper towel off. Change and back to the desk to cool off in stage 2 for another 20 mins or so. At this point I'm pretty much good to go.
I like the recommendation of cooling off for the last mile. My commute for the last mile is generally downhill, so this would work great, except I have to turn left across 6 lanes of traffic and there's a very short window of opportunity to take the lane usually. I tend to treat this as a field sprint, and that I'm Mark Cavendish (to the point of holding my arms out when I'm in the turn lane and pointing at my imaginary sponsors on my imaginary jersey, mostly to amuse my wife who's probably 20m behind me still cruising along ;-) ). So this tends to get my heart rate up, but it's fun and necessary, so cooling off the last mile doesn't work for me.
A fan might help cool off quicker. I used one for a while but I don't
really need it. So that's what I do, and it works well for me. Hope this helps.