On the RZ--any chance something metallic (non metered prism?) is making contact on the camera body where the AE viewfinder terminals are? Mine has a little plastic insert to cover those terminals up. Black electrical tape could work too. The contacts on the film backs are only for setting the film speed for the metered prisms. They shouldn't be doing anything to the shutter.
My waist level finder has no plastic cover for those contacts. I will try a piece of tape.
Strangely, the camera seems to work perfectly in the studio, even with 8 second exposures. It may have been the cold weather affecting the battery.