This plugin is amazing, let me start off by saying that. It's going to save me a lot of trouble making the game I've started making.

I've run into a problem when I try to export to GLTF where it ends up looking like the image on the left in Windows 3D viewer and also the program I'm using to make my game. I think it's a lighting issue because in 3D viewer I can make it return to normal if I change all the lighting intensities to 0. However I don't have control over this in my game's dev program. What can I do to fix this in Blender?

Can you share the glb file?

It would help for someone to load it into Blender to check if a parameter is missing during your export. Or to advise on one to tweak.

I can try to reproduce it and upload since I solved it by just going to shader and connecting the texture image as a background directly to the  material output. Though this probably won't work for other people because it makes lighting no longer work, which for me it doesn't matter since I have to do the lighting in the my game's engine separate from the file.

The issue is with your shader in blender. The default shader for  Sprytile does not import nicely into Godot. I suggest you make an actual Godot Shader. Even when I just used a principle bsdf shader in blender, it still doesn't import everything correctly into Godot. By creating my own custom Godot shader, I avoid that issue. When you import the gltf file into Godot 4.0 you can go to the materials section in import and override the material with your Godot material. If you are using some other engine, the advice remains the same, just create a material within that engine.