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You can get a shavette cheaply enough, which is the same as a cut-throat but uses half a safety razor blade. I had the same problem as you describe and switching to a shavette solved it. I do nick myself occasionally, but then I used to get cuts off modern safety razors too - usually when they intersected with ingrown hair bumps. I think I'm simply too hurried in the mornings.
For real? I want to believe, but don't want to spend lots of money on a quality razor and associated paraphernalia only to discover it's the same old story of rash, pustules and ingrown hairs.