Are you sure the blade is properly tightened? My Wilkinson requires some force to properly tighten, slightly bending the blade, at which point it's firm, sticks out less, at a different angle, and it's not as easy to cut myself.
Other helpful tips - let the hair soak for a couple of minutes; clean/dip the blade with >90% alcohol to remove moisture, preventing corrosion and dulling. Haven't had issues shaving since.
Honestly, you've got to be doing something wrong. Maybe a bad, non-lubricating shaving cream, way too much pressure, or inappropriate holding angle. Safety blades are not known for creating horror films.
If the motion of the blade is perpendicular to the edge of the blade, it shouldn't cut you no matter what angle you hold it at (some angles will cut your hair properly but none should cut your skin.) If the motion of the blade is parallel to the edge of the blade, then you're going to cut yourself.
It's not like other people have armored skin that's immune to sharp things; we all cut just as easily as you do. It's a matter of technique.
Have you tried various blades? People swore by feather but I found those too be too stiff and sharp. I absolutely hated them. I tried Astra Platinum and they seem more flexible and provide a much better shave for me. There's a "chart" out there that lists various brands from sharpest -> smoothest
Yes, I have, and yes, my latest have been Astoria, and yes, I pull, instead of slicing, and yes, I've futzed with the angle, and yes, I've tightened the razor, and yes, I've played with not tightening it as much, and the end result is that despite all this work, the disposable plastic crap is easier and less painful.
There's a reason for why the world has switched to the disposable plastic crap - and its not because people are fools. It's much the same reason for why the world switched from horses and buggies to automobiles.
With a cartridge, I don't cover myself in nicks. With a safety razor, every pimple, birthmark, and angle in my face is a potential cut point.