I’m a tablet user but not an Illustrator user; I use Photoshop. My advice, and I know that this isn’t what you want to hear, is that the problems you’re having will go away with time and practice. Seriously – I could not draw a straight line on my tablet when I first got it. For some people, it just takes a lot of getting used to.
It’s true that turning off or changing the pressure sensitivity would get rid of the width variations, but lines of variable (in a good way) width are something you’re eventually going to want to have, so rather than getting rid of width variation altogether, I think it’d be best to just keep practicing with it until you get control over it. Also, I don’t know if you have layers in Illustrator, but something that works for me is to have a rough sketch on one layer, and to go over it and refine the lines on another layer. Inking still doesn’t come easy to me after five(?) years using a tablet. I oftentimes have to go back and erase the edges of my lines where they are too thick to thin them down, or add more to lines that are too thin.
Do you have different brush sizes in Illustrator? In Photoshop, you could switch to a smaller brush to get thinner lines. Sorry I’m not familiar with the program you’re using.
Oh, and another thing. I don’t know if you have a setting called “jitter” in Illustrator, but in Photoshop, this is a term that basically means “randomness” and if it’s enabled with respect to your line thickness, it could be giving you a lot of unwanted results.