I have this problem. Wearing soft wool or cotton usually alleviates the problem. If I'm wearing anything synthetic it HAS TO BE FORM FITTING. Or else I get bloody nips.
And now everybody knows.
My summer garb is usually a tight sleeveless gym shirt with a loose fitting Columbia adventure longsleeve over it. The tight fitting shirt helps sweat form smaller beads and the long loose fitting shirt acts a wick and evaporator. This works even in hot humid FL summers! It also completely covers me from the sun and I don't have to use sunscreen anywhere on my upper body except face and neck. I'm a redhead and very aware of the dangers the sun poses.