Surface shader changing z value

Hello,

I want to make a surface shader and give bonus to its z values, eg make the illusion that its closer to the camera when doing ztest. For example, if 2 meshes are right onto each other, I want the mesh with my shader to hide the other mesh, because it has extra Z. If however its too far behind, it will be rendered behind.

I tried dding Offset attribute but it just doesn't work (not sure why) and it seems like its impossible to mix vertex shader with surface shader so I can't add a bonus to the output Z.

So how can I get the effect I want? Is it even possible?

Here's my shader (copied from Sprite/Diffuse of Unity) :

 Shader "Custom/SpriteShaderLights" 
 {
         Properties
         {
             [PerRendererData] _MainTex("Sprite Texture", 2D) = "white" {}
             _Color("Tint", Color) = (1,1,1,1)
             [MaterialToggle] PixelSnap("Pixel snap", Float) = 0
         }
 
         SubShader
         {
             Tags
             {
                 "Queue" = "Transparent"
                 "IgnoreProjector" = "True"
                 "RenderType" = "Transparent"
                 "PreviewType" = "Plane"
                 "CanUseSpriteAtlas" = "True"
             }
 
             Cull Off
             Lighting Off
             ZWrite Off
             Blend One OneMinusSrcAlpha
             Offset -10, -10        // <-- DOESN'T WORK, NO MATTER WHAT VALUES I GIVE.
 
             CGPROGRAM
 #pragma surface surf Lambert vertex:vert nofog keepalpha
 #pragma multi_compile _ PIXELSNAP_ON
 #pragma shader_feature ETC1_EXTERNAL_ALPHA
 
         sampler2D _MainTex;
         fixed4 _Color;
         sampler2D _AlphaTex;
 
         struct Input
         {
             float2 uv_MainTex;
             fixed4 color;
         };
 
         void vert(inout appdata_full v, out Input o)
         {
 #if defined(PIXELSNAP_ON)
             v.vertex = UnityPixelSnap(v.vertex);
 #endif
             UNITY_INITIALIZE_OUTPUT(Input, o);
             o.color = v.color * _Color;
         }
 
         fixed4 SampleSpriteTexture(float2 uv)
         {
             fixed4 color = tex2D(_MainTex, uv);
 
 #if ETC1_EXTERNAL_ALPHA
             color.a = tex2D(_AlphaTex, uv).r;
 #endif //ETC1_EXTERNAL_ALPHA
 
             return color;
         }
 
         void surf(Input IN, inout SurfaceOutput o)
         {
             fixed4 c = SampleSpriteTexture(IN.uv_MainTex) * IN.color;
             o.Albedo = c.rgb * c.a;
             o.Alpha = c.a;
         }
         ENDCG
         }
 
             Fallback "Transparent/VertexLit"
     }