If the light on the shifter is flashing it means that the power supply for the wheel is faulty and nto working. The shifter and pedals are powered through through the USB. The wheel is off the power supply. You wheel not alligning properly is the same issue as the wheel losing it centre when driving in games. Yours is just affect by it syncing. You have options, a) warranty claim if still in warranty period. OR b) Glue the optical wheel to the shaft with epoxy.(voids your warranty)
To sort the drivers there is a way to delete the old profiles for the wheel so that it makes a new one. Shouldn't really do that to be honest, most likely related to you unplugging it a lot (Maybe but i dont know). The driver when you first plug the g27 in is fine. Thats needed or else it wouldn't work. The logitech profiler is just to set custom profiles for steering angle, force feedback etc.
I had my first g27 for two months, the first 2 weeks it got the steering problem. Then in a race the power supply died and took me a 2 days to find out what was wrong with it. Sent it back to Boulanger (lived in france, now live in australia) My second g27 had the steering problem from day one so i glued it with epoxy rated to hold 140kg. Lets just say it does the job.