Remember although the shower may feel like a lot of waters comming through, its less than you think, and a mix of the two supplys. So running a HW tap is not a fair comparison.
Run a HW tap but slow it down quite a bit (not to a treakle, but maybe half way) and see if the boiler custs in and out, and more importantly if that is noticable at the tap.
If it is various issues could cause the boiler to do this. Most with Scale/Dirt, but some with Gas Regulation and electronics so do as above and if it is struggling keeping steady heat at the taps on mid/low flow, then its time to gall an RGI to look at the boiler