I had a similar problem yesterday with my 1999 323i. I turned on my car and the left turn signal would not turn off, I turned the wheel and it still would not shut off. I turned the signal to the right and the left signal would stop and the right would flash. As soon as the turn signal switch would return to neutral position - the left turn signal would start again. I searched the forum (E46 only) and was disappointed to find that no one else had posted helpful information regarding this issue. This gets a bit long - but the whole story must be told. Suffice to say - I fixed my problem by replacing my turn signal switch that is mounted on the steering collumn.
I had replaced the switch about a year ago for an unrelated symptom - where the turn signal would just become completely loose - but only occurred in cold weather. At that time I had taken apart that switch (after replacing it with one I purchased on line), and found that the spring plunger in the switch that holds it in position - was too big for the plastic hole that the plunger rests in. At cold temperatures (I live in MN) the hole the plunger fit in would shrink slightly, trapping the plunger. I verified this by throwing the now removed (old) switch in the freezer and testing it... similar behavior as to what i experienced when installed. I sanded the diameter down slightly - threw it in the freezer and tested it again - the small adjustment to the plunger size fixed the problem. I put the switch in a box in my shop, in case I needed it for another day. That "another day" was yesterday.
Yesterday I spent time looking through my Bentley manual and was not able to turn up anything meaningful on the light being stuck on. I tried the old switch that I had saved from my previous problem, and because I had reassembled the switch body wrong, it did not work at all. Still thinking I had a problem, I spent more time trying to evaluate if there was something wrong deeper in the car, a failed ground or faulty wiring. Finally, I removed the switch again and diassembled it, found I had put it together wrong, and reassembled it correctly. Plugged it in and now it works fine. I will be trouble shooting the faulty switch to see if I can see why it failed. If I find anything, I will post what I learn.
If you choose to replace the switch on your own this requires removal of the steering wheel and air bag. So please search the threads for proper Air Bag handling - this is a safety issue (obviously). So proceed at your own risk. I was able to remove the airbag, complete the removal and replacement of the switch - at least three times (because of my own mistakes) last night without deploying the air bag and also without throwing the Airbag fault light. The key is to remove the negative battery connection prior to unplugging the air bag from the steering wheel, and being careful to reassemble every thing prior to reconnecting the negative battery connection.