Hello,
You seem to be having hyperhidrosis. It may be primary (without known cause and usually starts during childhood or adolescence and persists all life) or secondary (as part of an underlying condition like Hyperthyroidism or similar endocrine diseases and dermatological conditions like eczema, psoriasis).
Treatment usually involves antiperspirants like aluminium chloride (20-25%) in 70-90% alcohol, Iontophoresis (applying low intensity electric to the palms and/or soles immersed in an electrolyte solution), botox treatment and surgery if none of the above methods are helpful. There are no specific drugs available for this.
Please consult a dermatologist to find out the confirmatory diagnosis as treatment depends on the same. It is very difficult to precisely confirm a diagnosis without examination and investigations and the answer is based on the medical information provided. For exact diagnosis, you are requested to consult your doctor. I sincerely hope that helps. Take care and please do keep me posted on how you are doing
In order to help yourself keep a cooler body temp you need to make sure you have a cold drink nearby as well as some damp washcloths or towels you can lay on your head to cool yourself off during your workout.
Due to my rare skin disorder I don't sweat well myself and tend to overheat to the point of passing out which can lead to death. I keep a spray bottle with cool water in it nearby to spray on my face and body so that as it evaporates it'll cool me off.