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Can't speak to a cut throat razor (that's what a straight razor is right?) but I moved to a classic safety razor due to price.
The cost of old fashion safety razor blades is so much cheaper than mach3 et al. OK I splashed out on a £27 razor but I could have bought the same one with a different grip for £20 and over the course of a year it paid for itself.
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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.
The way that I ended skin irritation and in growing hairs was by switching to a straight razor.
It's given me a nice hobby and made me feel 7% more manly, too.
It also carries a suicide abetting risk.