You are describing balanitis, eczema of the penis. This irritation canindeed be maddening, but is not infectious.
If I were you, I would try non-prescription (in the US) hydrocortisone 1%, twice-daily for 3-5 days. If that brings relief, it would tend to confirm the diagnosis. Armed with that information, I would then contact a physician for evaluation and perhaps a prescription for something a bit stronger.
I cannot explain why this problem should have begin just now. Nor is it certain that you will always have it. But while it's there, you should have something available to mitigate the symptoms simply and safely.
Best.
Dr. Rockoff
Please help