All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad is His slave and Messenger.
The matter of touching the Quran can be easily solved, as you can use something as a barrier to prevent the sweat from reaching the Quran such as wearing gloves or holding the papers with a tissue. As for calling people to Islam, it is not limited only to giving speeches; there are many other means to call people to Islam such as writing, distributing books and audio cassettes, and the like. If your sweat disturbs you during speeches then you may use the other means, and there is no need to worry.
We do not know any specific supplication in this concern but you may use the general supplications for seeking cure such as the supplication. “Allaahumma rabba an-naasi adhhiba al-ba’sa ashfi antha ash-shaafi la shifaa’a illa shifaa’uka shifaa’an la yughaadiru saqama” [O Allaah Lord of Mankind, remove the harm, and cure me for you are the Curer, there is no cure except Your cure, a cure which leaves no disease] [Al-Bukhari and Muslim]. You may consult the specialist doctors in this matter. See Fatwa 82537.