Newton and Einstein don't actually disagree with what
@Commander Biscuit said.
Both you and they are correct in your descriptions.
What's key here is 'relativity', i.e. how things appear relatively to different observers.
There's two key fundamental principles:
- The laws of physics apply equally to all observers
- The speed of light is a constant
(What that means is that any observer will measure the speed of light to be the same, no matter what their reference frame is.)
For someone on the ship, it doesn't matter what speed they're travelling at, the laws of physics apply as normal, and if a constant thrust is applied, they will constantly accelerate, and to them they will appear to be doing so according to the standard laws of physics. (Essentially, an accelerometer in the ship will measure that they are accelerating according to the thrust exactly the same as if they started out stationary or at any other speed.)
For someone outside the ship, who the ship is accelerating relative to, the ship will indeed appear to accelerate towards the speed of light at an ever decreasing rate.
Einstein's relativity is essentially understanding working out how things will appear to different observers in different reference frames, and what that means.