How would I recreate a building tool that can place blocks?

One reason it might be is because you are using Mouse.X , Mouse.Y, and Mouse.Z. To fix the problem with the Z, is to change each of them to Mouse.Hit.p.X, Mouse.Hit.p.Y, and Mouse.Hit.p.Z. Or otherwise, you can just erase X, Y, and Z and use Mouse.Hit.p for the whole set of 3 arguments, since Mouse.Hit.p returns a Vector3 Value.

So now, for the whole Drag’n’Drop the pieces system, you can create a new variable, called Target. Then you can create code to set the Target to mouse.Target when the mouse presses down on an object, and then sets Target to nil when releasing. Simply using mouse.Target doesn’t work, since eventually the player might move their mouse too fast, and end up dropping the piece. Using a variable, will make the piece the same one the player starts dragging, and Target will stay until the player lets go of their mouse, so from a LocalScript, you could try this:
local player = game.Players.LocalPlayer
local mouse = player:GetMouse()
local target = nil
mouse.Button1Down:Connect(function()
if mouse.Target then
if mouse.Target.Locked == false and mouse.Target.Anchored == false then
target = Mouse.Target
end
end
end)
mouse.Button1Up:Connect(function()
if target then
target = nil
end
end)
while true do
target.Position = Mouse.Hit.p
end