I too have a 700 MLS with a weather shroud. I close my bolt and put a line with a pencil on the bolt in the receiver then install the shroud so the hole is lined up with the pencil line (it should be just above the receiver in the closed position to allow smoke to escape) If it is really wet, I will replace the primer often and dry the nipple. Carry my capper in a sealed ziploc. Another thing I found that works well is a wool sock over the breech area if stand hunting, replace as it becomes saturated.

If it is wet, snow, high humidity, etc I always tape the barrel with electrical tape, it does nothing to the POI or Ballistics. The gas will blow the tape before the bullet. I took 3 animals with the taped barrel, no problem the bullet went where it was suppose to.

I install the breech plug & nipple with teflon tape (pink plumber style) to prevent moisture from the breech area.

Whatever you do at the end of the day fire it and clean it very well when it is damp outside.