If the lens is sticky and the cocking ring pins that mate to the camera body, do not fully return to their open position, then the camera refuses to advance.
Try dismounting the lens and pushing the pins to seat them fully open (check mating ring as well and turn to full open), remount the lens and then see if the camera will advance. if this proves to be the case, you might be able to exercise the lens shutter off of the camera body by manually cocking the shutter with the pin ring and pressing the small release pin on the edge of the mount. You might have to push the ring through it's range, but that won't hurt the lens.
I think you are on the right path; try getting the lens cocking mechanism serviced and hopefully you won't have to have the lenses serviced as well. It is a combination of the spring power of the camera body and the lens shutter release mechanism that returns the plate to the proper position. If either one is weak, it may jam the camera.