Having a secondary means of transportation is a good option. When the weather gets too hot or the air quality gets too bad I will take the bus to work (nice and air conditioned) and then ride home so I could care less about how much I sweat and I get to end the ride with a nice, cool shower.

Could you take a bus? Is there a way you could rent a scooter at a reasonable price just during the months that it is sweltering? How about an electric-assisted bicycle (you still pedal, but you are going much faster with less effort)?

It rarely gets as hot here as it does in your neck of the woods, but there are days where heat and humidity make sweating nearly impossible to avoid. The worst part is have to sit in your own sweat all day...not fun. I don't have a shower at work either and have tried most of the same things (except the frozen t-shirt). The rack/pannier solution should keep you a little cooler, especially if you are currently using some kind of backpack or messenger bag. If you are able to take a cool shower at the sauna in your office building each morning I think that would make the biggest difference.