We want to improve the insulation in a bedroom. Will it cause a problem (condensation, etc) if we put rigid sheet foam insulation that completely covers all the studs before installing the sheetrock? There will be fiberglass insulation between all the studs.
In Oregon you are primarily a heating climate. The typically thing to do is to have a vapor barrier on the warm side. Normally you insulate between the studs and then put 6mil poly and then drywall. The moisture in the air will then not hit a cold surface as your drywall is on the warm side and the air can't get to the cold exterior sheathing.
Now as far as your plan to use XPS - high density foam - on top of the studs before the drywall should be fine. The XPS is a vapor barrier but you'd need to tape the seams or have poly in front of it. Boarding will be more of a challenge though as you can't see the studs.
I don't think the screw heads are any more likely to cause a problem than the walls. If you have high RH levels you should have a dehumidifier.
Condensation on walls occurs when there is high indoor humidity and the walls are cold (insulation is bad and the heat is either low or not adequate to heat the surface of the wall). I don't see insulating / air sealing / putting up a vapor barrier making any of those things worse.
We live in the same area, you will want something that will allow moisture out But stop air flow (vapor barrier).
We do have issues with mold here. on several houses I have found the bad stuff (Stachybotrys chartarum) and 2 have been when the home owner tried to seal the walls In similar era homes. In this era home there are normally a lot of air leaks and house wrap may be a better option, although toxic mold is not common all mold can cause issues with the structure and health at some level.
Also don’t forget about your floor. Your floor is a large external surface area and simple fiberglass insulation in both the walls and floor may be best. I have used chicken wire or fish net to hold the fiberglass in the floors in place and it reduces the chance of squirrels, raccoons and opossums getting in there and pulling it down (yes even in town). if copper pipe do not let the chicken wire touch the pipe, the moisture in the air + copper+ zinc coated wire = pinholes in copper pipe because of galvanic corrosion.
If you want to air seal the room please get a dehumidifier. Dehumidifiers in a sealed bedroom will go a long way to prevent mold and mildew here in the Pacific Northwest.
I don't think you can attach drywall over (flexible) plastic foam. When my house was built I put in the wall insulation; I put the paper-facing of the fiberglass batts ( flanges) on the 2 X 4 wall studs. The dry wall guys said not to put anything on the 2 X 4 surfaces , they wanted to place the dry wall directly on the bare wood. The dry-wall installers were specialists that only installed dry-wall.