One of the interesting observations I've had on the bike is that when it is hot and muggy out I don't notice the sweat until I stop. I can be keeping up a decent pace for 5 miles, and it's only when I have to wait for traffic at a crossing or when I arrive at the destination that I'm suddenly drenched in sweat.
The body can keep sweating for quite some time after exercise if it's still having to get rid of some excess heat, often I'll get home from a ride, shower and still feel like I'm (slightly) sweating after.
I'm the same way. Sometimes I'll try to ride slowly to avoid getting sweaty, but I'm rarely successful. I'm just used to riding faster, and it's more fun. As you said, it also can feel safer/more natural if you're next to car traffic. Luckily I've usually worked places that had showers if I really needed them.