Hey, I came in here looking for help with this topic (which thanks again everyone, helping someone else end their nightmare). Fusion 360 does actually let you save stuff offline, it’s just a different way of doing so for the drawing files; you do a “Save as” file, and you can choose where it goes.
as for saving .stl files in fusion 360, you do “File” → “3D print” instead of the “make” button. A dialog box appears that has a set of drop downs. The two important ones here are the ones that say “refinement options” and “output”.
Immediately uncheck the box that says “Send to 3D print Utility” under “output”, and this will do two things. The first thing is it will unlock two greyed out boxes under “Refinement options” and the second is it will let you save your .stl to your computer. As for the refinement options, the defaults should be good, though I recommend setting “surface deviation” to .0002mm and maximum edge length to .005mm; this will result in a .stl file that is gorgeous, but not needlessly oversized.