Yes, the theory of relativity is unintuitive, but it is both internally consistent and there is loads of evidence for it. Your example, Sean, is no different. If you would sit down and do the maths (which is not trivial, of course), you would see that everything works out. I mean you don't have to believe me, but you shouldn't doubt everything scientists say just because it seems unintuitive to you, either.
But in principle, what you have to do is to start from the constancy of the speed of light in any frame of reference. This is itself paradoxical in classical physics. What you will find though that if you then assume length contraction and time dilation, things work out. Essentially, you start with the constancy of light (and the equivalence of frames of reference) and *derive* all these weird effects from there.
In your example, well the first question: One light second of length in whose frame of reference? Let's say in your own (where you are the guy who is attached to the laser).
So what happens in your own frame of reference? Well, you shoot the laser, it travels the distance to the target and hits it after one second.
What about the frame of reference of an outside observer, relative to whom you travel with almost c in the direction of the laser beam? Well, he would see you with almost c. He then would see a laser beam emerging from you (well technically you can't see the light till it hits you, but bear with me), travelling at c. That means in his frame of reference, the 'tip' of the beam would move away from you, its source, only very slowly. Thus, in classical physics, if you were holding a one light second long target ahead of you, say attached to a stick, it would take much longer than one second to reach the target. So how can this be?
Well, what I just have said is that it should take longer than one second to hit the target. And it does, for the outside observer! And that's time dilation. What takes one second in the moving frame of reference takes longer in the stationary one. The duration laser fires<->target is being hit is one second for you, but from the observer's point of view it would take much longer.
So in your example, time dilation would offer a solution for the apparent paradox that in your frame of reference, the light travels away with c, but seen from the outside observer, the light still travels with c, but then increases distance from its source much slower.
Alternatively, length contraction is another solution. Yes, the light travels away from you much slower (in his point of view), but the distance to the target, i.e. the length of the stick, is much shorter because of length contraction.
In reality, it's a wild mix of time dilation, length contraction, and a change of simultaneity (events that happen at the same time in one frame don't in another) that will make things come out exactly right. And again, the way you have to think is not that all these weird effects conspire to magically fix the problems coincidentally just in the right way, but rather, that when you assume equivalence and constancy of the speed of light and allow for time and space to be non-constant, then you can *derive* these effects (with the help of some further constraints).
Edit: typo, minor clarification