I don't know if I'm explaining exactly right so any reccomendations for title improvements is welcome
Is there any shortcut that I can drag out a new vertice like this, but still maintain the old one?
The Rip Fill feature in the answer by Paul is the one you want. https://blender.stackexchange.com/a/91703/13643
These are just alternative methods you can try:
Connect those 2 vertices and press J to Vertex Connect Path, then select the faces with extra edges and use AltJ to execute Tris to Quads so it dissolves the unnecessary edges.
Connect those 2 vertices using Knife Tool K and then use Tris to Quads AltJ as in the previous method:
Or you can select 2 vertices and make a new vertex between them by W > Subdivide and turn on Quad/Tri mode. Then again, select the faces and use Tris to Quads AltJ (or just use this at the very end on the whole model).