If your lines are breaking there is too much movement between keyframes. The Brush Spacing isn't in distance, it's in time. IOW, if your brush moves more than it's diameter between frames there will be space between the dots. The minimum spacing is .001 seconds. That means that if the brush moves more than (1 / frame rate) * 100 * half the diameter of the brush you'll start loosing the integrity of the line and the edges will scallop. At 4 times that distance per frame you'll get dots.
IOW if your brush size is 10 pixels the maximum movement between keyframes would be something like 16 pixels. (1/30 * 100 * 5 = 16.6666) You could do the math if you'd like, but all you have to do to fix the problem is to give a little more time between keyframes.
As Rick said, simply increase the time between keys. The simplest way to do that would be to select the keyframes, then hold down the Alt key while dragging. This will scale them temporally. If, for some reason you want to keep the timing as it is, pre-compose the Write-On part, then increase the composition's framerate to multiples of that of the parent comp, e.g. if the main comp has 30fps, you can use 90fps in the pre-comp. that way, you are introducing additional inbetween steps that force AE to evaluate the motion more finely. Of course the underlying physics don't change and there are limits: If your effect is still moving too fast even for 90fps, you'll get broken lines again, so probably a combination of both techniques will give the best compromise.