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Your formula is wrong.

    Gravity above Earth's surface is g' =  G Me / (r+h)² 
     Force of gravity = m g' 

See the diagram enclosed.

   When the mass m is below earth's surface at a depth of h , then there are two forces acting on it.  One is the gravitational force of Earth inside radius (R-h) and another, the gravity due to Shell of thickness h.

The net force of attraction of the outer shell of thickness h is zero. This is Because of symmetry and direction of force being  spread in 360 degrees from all sides.

Density of Earth, = Me / (4πR³/3) 
Mass of Earth with in radius (R-h) =   Me * [ 4π(R-h)³/3 ] / (4πR³/3)
                                        Me' = Me (R-h)³ / R³

Hence the gravity (acceleration)  g' below Earth's surface =

        g' = G Me' / (R-h)² = G Me (R-h) / R³
           = ( GMe / R² )  [ 1 -  h / R ] 

        g'   =  g [ 1 - h/R ]
       m g' = m g ( 1 - h/R )