An elderly Italian barber taught me about shaving. According to him, there isn’t anything intrinsically magical about using a brush or a straight razor or any of the other lovely old-style accoutrments. The secret to a great shave is to properly prep your beard before you shave. The brush helps work the lather into your beard and the lather gets a chance to penetrate the whiskers while you’re brushing it in. The shaving soap lather softens the beard and provides lubrication, the stuff out of spray cans does neither. He also emphasized that irritation is minimized by shaving with as few strokes as possible.
I was introduced to wetshaving when a guest named Leisureguy had a comment about a staff report. that went pretty much unanswered. His shave site is awesome though. I suggest giving it a read! I have yet to migrate to a safety razor, but using the other wet shave techniques have turned a chore into a spa-like experience. I love shaving now!
I forgot to mention that I shave after showering. If I don’t shower, I don’t shave. My beard becomes quite soft and there’s relatively little irritation.
(I tried shaving in the shower but I couldn’t get every whisker. Also, it seemed to take longer.)
I think that it does both, though perhaps not as well.
I tried using a brush and shaving soap, and it didn’t seem to make a difference.
Maybe it doesn’t matter much if your skin is not very sensitive.
It’s hard to say what specific components have made shaving far less irritating for me, since I made so many of the changes at once, including using a good technique. No goo or cream from a gel feels anything like the super lather I make from real shave soap and cream. It’s so slick and luxurious, I can’t explain it. It’s one of those things you have to try for yourself to enjoy. Plus, it feels sooo good going on with the badger brush- it’s very relaxing. I think it’s a more change important than switching to double edge. The glide I get makes my skin much less prone to irritation. Going back to canned stuff after using the real deal, and it almost feels sticky in comparison.
I believe having only one blade go across my face at a time instead of two or more also helps with irritation. Plus, old school blades are just so much cooler- and cheaper! Bu most of the stuff still available on the drug store shelves are garbage. The blades on the internet are best. Some recommend getting a sampler pack like the one here to see what works best for your face. Watch out for the Feathers, they aren’t called Ninja sharp for nothing! Not a beginners blade. My friends and I agree that Derbys are best ( I got plenty to make the switch 
), but you’ll get all sorts of opinions on the various shaving forums. I don’t have to go against the grain anymore to get a smooth shave. I make one pass going down, not worrying about getting everything. Rinse. Re-apply lather. Go across the grain. Rinse, Check for missed spots. Rinse off razor, rinse and shake out brush. Done. One cold water rinse of the face and a slap of Lumene (only available at CVS) or Nivea Extra Soothing Balm and I’m a new man! 
ETA: Don’t forget to stretch the skin, especially on the final pass- very important. It helps to get as smooth as possible without going against the grain.
I second the shaving in the shower. I use a badger brush which I lather with straight bar soap.
Like yourself, I would cut myself regularly and still do when I must shave over a sink!
I’m impressed with Neutrogena’s Shave Gel and Triple Defense Face Lotion.
I’m a girl with exceedingly sensitive skin. Of course I don’t have the shaving issue, but I feel your pain. Literally. I can recommend Origins Constant Comforter and Aveeno Ultra-Calming Night Cream (which you can use during the day).