I preheat my mash tun as well. The only difference is I dump the preheat water seconds prior to adding my mash water. Then I add my mash water, then the grain (while stirring), get the temp right, close it up cover it with a blanket and a space blanket,and check temp and stir every 15 minutes.
I have a digital thermometer that runs into tun that always displays the mash temp but I also put a glass thermometer in the grain bed too verify the digital temp.
Personally speaking, I always have hot or cold water ready each time I open so that I can add if necessary.
If you are using a Rubbermaid cooler, I suggest insulating the lid. I drop maybe 3 degrees an hour.