This sounds a lot like an accentuated normal reaction. What I mean to say is that many people, when they are anxious, stressed, or otherwise in a state of heightened emotions, tend to have some physical symptoms, one of the most common of which is cold and sweaty palms. This is actually a totally normal reaction to the stress of the situation, provoked by the part of your nervous system which deals with stress reactions. There is no need to treat it under most circumstances.
Some people will have an exaggerated reaction in these situations and the sweating and other symptoms can be socially embarrassing. In these situations only do doctors
tend to treat the symptoms. As you have already experienced, the most common treatment is a drying agent like aluminum chloride. One of the side effects of this treatment is that it tends to make the hands quite dry. However, there are no medical treatments that do not have side effects, and the side effect of Drysol is very mild.
Your only other options are to try to regulate your stress levels, by breathing exercises, and maintaining a good diet and exercise plan. If you have further questions, talk to your primary care doctor