After a period of 15 years break from cycling i am now back in the fold and enjoying more than ever. Even when i was younger the fitter i got the more sweat poured and i mean poured out of me. I feel fitter now than ever in 15 years but the issue is i sweat within about 10 seconds of doing the mundane chores in life. It is a strange scenario as many think i am unfit but they could not be more wrong. I helped out lift a chair just 10 steps and i was sweating so that i had to change. I have had a full medical and all is fine.
I know some sweat more than others but its getting an issue. I had this when i was young and so its not new but it seems worse now.

Do others seem to find the fitter they get the more sweat they produce

As a WAG, more muscles = more active = more sweat? As you get fitter, you move more, generating more heat. Instead of sitting in front of the TV or computer, you're walking, cycling, etc. Your body dumps the heat by sweating. Because you're actually doing something, instead of passively absorbing photons, you notice the sweat.

I read a runner's article sometime and it mentioned this phenomenon. For folks that do a lot of running and exercise, your body is used to activities that require quick heat rejection. Your body starts to think that as soon as you get an uptick in heart rate or activity, you are going for a long run. So it starts dumping out heat and making you sweat right away.

This could be what you are experiencing.

Try taking about 1 gram of sage leaf twice a day, and see if that helps. You can find it in capsules or just grow it.

I find that the more fit I am, the less clothing I wear. Additionally, it takes me forever to cool down. I'll be fine riding somewhere and not even notice that I'm sweating much. I'll go to lock up my bike, and sweat will just keep pouring off of me.