Hi, sorry for the long inactivity.
In my post below, I show how to embed textures into a 3D format that allows image embedding such as the FBX format in Blender:
Please note that, “solid colors,” which I assume you mean, Vertex Colors in this context; instead of Object Color or Material Color, image textures can be exported if the Blender exporter supports it (PNG, JPG, DDS, etc). Additionally, note that you also need to properly setup your 3D model’s Material Nodes to correctly point towards your used textures.