OK Yeah, I'd probably keep that map as a digital file, and put big Xs on it when the PCs have cleared a room. I'd probably keep a separate doc with the rooms the PCs have visited, listed out with any notes about what happened there.
Since there's a detailed map, I'd probably never change around the layout of the rooms, that would just be annoying to keep track of. However, I'd feel free to change what's in them in terms of treasure and enemies, depending on what made sense at the time. Note that in the doc as you go.
If you're feeling ambitious or are familiar with an image editing program, you could have a PC version of the map, which you send them after each session, removing the fog of war from the parts they haven't explored yet. They probably need to see the map that they've already explored, to stay oriented.
I probably would read ahead for the ring of rooms nearest where the PCs are. They can always take a totally different tack, but might as well be prepared for the most likely one.