New float valve in header tank with very slow leak
The high water pressure could very well be causing the slow leak. Float valves rely on the pressure of the float to close a small valve in the stem. High pressure could allow for a little water to work its way through. A quality fill valve should be able to handle it. The fluidmaster 400 ARHR is a great low cost fill valve. You could also try a commercial fill valve. They are built to higher standards.