You calculate the acceleration in an inertial frame (supposedly the frame at which the spaceship was launched). If the acceleration is constant 10m/s² as seen from that frame, the acceleration experienced by the observer on the spaceship would be much higher once it reaches relativistic velocities.
Also you only calculate time dilation experienced after turning off engines, ignoring time dilation during the acceleration.
You make the correct observation that you cannot maintain constant acceleration (as seen in an inertial frame) indefinitely, because you hit the speed of light. But you can maintain constant acceleration in the accelerating frame indefinitely. So from the point of view of the rocket it can keep accelerating 10m/s² until the midpoint, and then start deceleration. And that's the fastest survivable way to get anywhere.
To get correct results you'd need to calculate that in Rindler metric: https://en.m.wikipedia.org/wiki/Rindler_coordinates
Sorry, my math is too rusty to come up with the valid calculation right now :)
Here is the pretty good course on relativity, Rindler metric was explained in first or second lecture: https://www.coursera.org/learn/general-relativity