So, if time slows down as one approaches the speed of light
This statement doesn't make any sense. What would it mean for you to approach the speed of light? Relativity means that your speed according to your perspective.
You will never measure yourself going near the speed of light, and there will never be an experiment that you can do that will show you going near the speed of light.
Time dilation happens when you compare what is happening in another frame of reference compared to what is happening in your frame of reference. If these two frames of reference are different, you will measure time going slower in the other frame of reference
Let's take the spaceship example where you on spaceship A notice that the distance between you and spaceship B is rapidly decreasing. You interpret this as you being stopped and the other spaceship moving rapidly toward you.
If you can measure the rate that a second hand moves on a clock on spaceship B, and compare it to a clock sitting next to you on spaceship A... you will say correctly that the clock on spaceship B is ticking slower (i.e. when your clock clicks 60 time, the other clock will click fewer times).
Of course on your spaceship... the clock always clicks once every second because the definition of a second is how often a clock clicks.
So saying that "your spaceship is approaching the speed of light" doesn't make any sense.
What you could say is "your spaceship is approaching the speed of light as measured by someone on spaceship B"? Of course in your perspective it is spaceship B that is approaching the speed of light... since from your perspective you aren't moving.