In a Newtonian sense, you're right. But black holes are not Newtonian objects, and although you get the right answer for the Schwarzschild radius, it's completely by incorrect reasoning. It's not actually an issue with special relativity -- but with the much larger theory of General Relativity.
If you imagine yourself going towards the event horizon of a real Einsteinian black hole, you would need to accelerate infinitely fast as you fell down towards the event horizon just to stay at the same radius
! Curiously, once you've fallen past the event horizon, any amount of blasting your rockets you do will only hasten your eventual demise! If you want to live as long as possible, the best bet is to sit there and do nothing.