If not too late ...
Had a similar, but somewhat different problem.
I didn't have the faulty lamp warning, just all the left side signals stopped working. So not a bulb issue.
Dealer diagnosed it as the switch and did a warranty replacement.
This was the part no.:
Bimmer Specialist
Not a common X3 issue, but I saw a few E46 posts on it.
With your symptoms, such as working fine with the engine off, then more bulbs coming on than are meant to, I would first suspect the classic problem of a faulty ground somewhere, or a broken filament of a double filament bulb.
Other than that, keep in mind that the indicator stalk is multi-function, including selection of OBC displays. These are active only when the engine is running, so a faulty contact there could conceivably cause strange interaction with the turn-signal lights.
Might be worth trying to spray some contact/switch cleaner in the part of the switch that sits inside the steering column to avoid the $130 replacement cost.