MakerBot Print is our newest print-prepration software, which supports native CAD files and STL assemblies,
allows you to interact with all your printers via the Cloud, and many other exciting new features.

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As nearly as I can tell, it looks like Tinkercad will allow me to export "For 3D Print" to .obj and .stl, and "For Lasercutting" to .svg. Probably the best thing I could offer is to make the design public on Tinkercad and see if you have any more ability to futz on the Tinkercad platform itself. I've never tried to make a design public on Tinkercad nor have I tried to mod someone else's Tinkercad design, but with some quick Google-fu, it looks like exporting a "futzable" drawing from Tinkercad into something else like Fusion360 isn't really an option. So if Tinkercad does actually allow someone to mod someone else's Tinkercad drawing, then that's probably your only real option to make any meaningful changes. Hopefully, some more knowledgeable user will chime in and prove me wrong.