Beammeup, because your reel blades seem to be running some distance away from your bedknife, you can at least fix your folding-without-cutting-the-grass problem by adjustment. Try a sheet of newspaper between the reel and bedknife, and turn the blades by hand (spark plug lead disconnected, of course). If, as I expect, the newspaper is not touched at all, you could practice adjusting it - on both sides individually of course - until it cuts or tears the newspaper most of the way across, but doesn't touch the fixed and moving blades against each other. (You can't expect scissor-like paper cutting from blunt blades.)
You said the reel only turns half a turn or so, despite there being clearance between fixed and moving blades. This sounds as if your reel bearings may not be in great shape.
The reason I suggest you practice a bit more before sharpening the blades is that if you make a small error in adjustment and there is light contact between them, it won't really matter because they need sharpening anyway.
