So I decided to just create a model by just putting in the "Add Locations" tool. I then made the following model parameters: Input Network Analysis Layer, Input Locations, Sub Layer, and Field Mappings. The problem is no longer there.
So after comparing this to my original model, I noticed that the difference was that in my original, the network analysis layer (Service Layer Name in my original) was not a model parameter. After making this a model parameter, I no longer have the issue. I have included screenshots below of Add Locations by itself, my model in model builder, and then the resulting model once run from a toolbox. So far it works! Thanks again!