There is more than one technique used to detect a "wobble".
One technique, as you point out, uses amazingly small red and blue shifts to determine when a star moves further from or closer to the observer. You are correct that in that case, we have no information to deduce the orbital plane.
Another technique (usable only on closer stars, but as instruments become more sensitive, the range increases) actually measures the apparant movement of a star. Usint that method, we could have geometric information sufficient to deduce orbital plane.