This might not be the answer to the question you've actually asked, but there's too much to put in a comment, so I'm going to risk the downvotes.
Your question suggests that you believe the sweating is being caused by being overweight and thus heating up faster, causing your body to produce sweat as a cooling mechanism. While this may well be the case, temperature regulation isn't the only reason we sweat, it also helps the body eliminate heavy metals and some chemicals.
Anecdotally, I've found that my body tends to sweat more for a number of reasons, not directly involving exercise. If I have a couple of nights of bad sleep, I don't keep myself adequately hydrated, my diet goes a bit wonky (think, less vegetables, more package food), I notice myself sweating more.
This is somewhat supported by some Googling around the subject.
Disrupted sleep leads to a disruption of thermoregulation, which could cause you to sweat more as a compensatory mechanism. Likewise, ingesting more chemically laden food may also cause a disruption in thermoregulation as well as just causing your body to try and purge the chemicals it doesn't have any use for (I don't have any links for that, it's partly more a gut feel type thing and partly because I can't access some of the links on the network I'm on at the moment).
So, instead of looking at exercise and "getting fitter" as a way to reduce sweating, it might be worth looking at your sleep and diet instead.
This approach can have the added benefit of helping lose fat; to quote Josh Hillis
- How much you weigh is determined by how many calories you eat.
- Your bodyfat percentage is determined by the quality of the food you eat.
That second point is the important one here.
Plus, the lighter you are, the easier you should find it to climb stairs :)