I’m pretty sure that Blender already has this. I’m almost certain that it was added fairly recently and that I’ve used it before.
Yes unfortunately is there just for edge slide, if you grab/move or slide a single vert the option “correct uv” isn’t there.
There is example scene
There is an addon by the name of moveuv by kgeogeo (He is on the forum, contact him directly for the update) it lets you move vertices.
By the way, most of the times, edge slide with correct uvs is the best option. one rarely need to move the vertice for better uv.
i’,m not sure off the setting off the top of my head but this is definitely an option i have used before. you can also add in extra geometry as well. i think its a tick box in the uv settings.
blenders busy so i’m not going to research it any deeper today.