A couple of things first...
1) Time dilation is not a consequence of high speeds, but of ANY speed - it just the effects grow large rapidly within about 10% the speed of light. Low speeds can have measurable consequences as is the case with magnetic fields for example.
2) All identical clocks "tick" away at the same rate under all circumstances,** metering out their proper-time distances, so be careful using expressions like "time slowing down" as if "time" were a thing itself that moved or flowed. This will help keep in mind that "time" is component, like space, used to locate a point in spacetime. It's a subtle distinction but one worth bearing in mind.
**Clocks in relative motion and various points in a gravitational field move along worldlines that are not necessarily parallel and thus project different time intervals (coordinate times) onto the other clock's worldlines. This effect is time dilation.
That said, your "How correct is" part of your question is not too far off. That is, the greater the motion of some other object through your space (greater spatial component) the less the other object moves through your time (smaller temporal component) because the spacetime interval ($ds^2 = dt^2 - dx^2$) is an invariant quantity - is what I think you meant to say.
If you haven't done so already I recommend reading Sofia's comment as she works through the arithmetic and this may aid your understanding.