A couple more tips....
You can also work up to it, practice doing a half a turn until you feel pretty comfortable, and everything is aligned and centered like cdkc said, do a number of 3/4 turns, then a whole turn, then 1 1/2 turns.
Another thing is too keep your head level, a lot of novice dancers tend to look up or down at the floor. This will make it even worse. In Victorian times, for a cheap high while V. Waltzing, dancers would throw their head back and watch the ceiling.
But I think to get over your phobia, you need to learn to do 2 things. 1) Enjoy being dizzy. It means you are alive and dancing. Like embarrassment, you aren't going to die of it. 2) Learn how to recover faster, so that you feel like you are in control. There are couple of tricks that I have learned. This one I discovered myself, when dizzy, focus on something that you know isn't moving, eye-level and vertical. A door frame, a picture frame, a pillar. I found that this gave my mind a reference point, and allow it to put everything else around it in their places. The other trick I learned from a dance instructor, turn the other way once. This counteracts the movement of the liquid in the ear canal which is typically what makes you dizzy.
But otherwise realize that few of us are good spinners, I still remember the day I had the breakthrough and got complemented on by my instructor, after have been on that plateau for months of my turns not being good enough.
-TheDitz