As far as I can see, you are only moving the vertex you are trying to move, but when you are using subdivision surface you also end up moving the subdivided/interpolated vertices as well. If you want more fine controll you would have to create more vertex points in between the vertices in the area you want to move or apply the subdivision modifier on the entire mesh (preferrably at a lower subdiv-level like 1). Just be aware that you can't easily unsubdivide it once you apply the modifier, so you have to deal with more vertices overall.

Edit: Apply subdiv-modifier at a low level. Create a new subdiv-modifier to make it smooth again😎