I don't think you'll need to upgrade your router hardware in order to make 3D parts. It sounds like you've been using 2D software, so you've been getting 2D results. Try downloading some 3d CAD and CAM software and see if there's any reason you can't cut in 3D. A demo program would work to show you what's possible. Try downloading a copy of Rhino from Modeling tools for designers, make a 3D part by creating some curves and creating a surface between them, save it as an STL file, download the demo copy of DeskProto from DeskProto offers rapid prototyping using an affordable cnc milling machine: DeskProto is 3D cam software for model making. , pick the machine that's closest to yours, and use the 3D wizard to create a toolpath. My guess is that the 3D carving will work just fine.
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