Hi Elad,
indeed FEFLOW adds water to the drying elements in case of 'unconstrained'. This is done by internally setting 1st kind boundary conditions at the corresponding nodes with h = z_bottom + residual water depth. You can analyze these boundary fluxes in the budget analyzer by using the buttons in the line in white characters at the bottom of the budget analyzer window.
The only possibility to avoid these additional injections of water is to use 'unconstrained'. The transmissivity for the dry areas is then calculated by residual water depth * conductivity. So using a low residual water depth, you might not have to lower the conductivity at all.
Cheers,
Peter