Ella's post, is very good. I would try the clinical strength anti-perspirants, from the store. They are not as strong, as a prescription, but, not as harsh on the skin, either. Making sure your bra, is always freshly laundered, that is important. When I went to America last year, I packed very light, I only brought three bras with me. I would wear the same bra, two days in a row, at times. As I biked everywhere, I would sweat. I realised, it was causing me, to smell, even when I was clean. Baby powder, could be of use to you, also.
I would be careful, about taking zinc supplements. Many people are sensitive to zinc. It can cause quite unpleasant stomach upset. A multivitamin is likely to be fine, as it will not have an excessive amount of zinc. Taking it separately, in larger amounts, is really not a good idea, unless your doctor has done blood work, and told you that you have a zinc deficiency.