Maybe throw them in for one of the cycles with detergent only? It might be detergent build up on the covers. Synthetics are more susceptible to build-up, which is probably why it's not happening to your flats. You might want to hand wash your covers with a squirt of dish soap like Dawn once (make sure to rinse them clean!), and then just go back to normal, but I think maybe just throw them in the wash after your cold wash cycle.
If the ammonia smell is really bad, I think there's some sort of fish tank stuff that people use to get it out (one time use sort of thing)? I'm not sure.