It sounds as though the clayey topsoil overlies a clay subsoil. The clay subsoil will act as an impermeable barrier for water and the water table that you describe is more like a perched water table.
The French drains that you have described should work fine as long as it drains freely away from your property, and make a noticeable difference.
Also, as you say that your garden is in a dip, then look for ways to prevent surface water runoff from entering your property.