For anyone who runs into this same problem that I created, I realized that when editing a Cad Detail, you can also set the Line Weight Options for Cad Details in the Drawing Sheet Setup dialog under File->Print->Drawing Sheet Setup. When I changed the Line Weights there to match my Plan and Layout files, it fixed the Line Weights on my schedules and everything looks correct now.
wazzubrad's post in Schedules - How to adjust Line Weight of "3D Exterior View" column? was marked as the answer