In addition to the straight razors, there are also safety razors that use a stainless steel double-sided razor blade. A friend of mine, also with sensitive skin, got one and loves it. It's a whole ritual with the badger-hair brush, building up the lather, and so forth. If you don't want to go electric, that's what I'd try. One of many brands you can Google for is the "Merkur Futur".