Random misfiring/rough running, no code

Car started misfiring/skipping randomly. When it is not doing this, it runs, idles, and cranks great. Started out, just a mild skip, but keeps going. After a while, it got worse and would skip and misfire a lot and backfire under the hood. It would even shut off, but would crank back within a few seconds. When it does this, it takes a few minutes to get it to go again without shutting off. But most of the time, it will skip a few times, I let off the gas a second and then it quits and I can go. Repairs so far: Codes p0404, p0401. Replaced EGR valve and connector due to wires pulled out. No change. Mechanic got random misfire code, but did not tell which cylinder. Had codes checked at the parts store, only p0420. Took to a different mechanic, said needs catalytic converter and o2 sensors. Did both. Drove good about an hour, then same symptoms. He said the checked the coils and they are good. He also said both motor mounts are broke, though we just replaced both within two months. Said next should replace the Ignition Control Module. Had a new one that was sitting in a none running car, it tested good at the parts store, so husband put it in. Did not make it a mile before it quit. Finally started back and got it back home, but kept shutting off. Tested the ICM we took off, and it tested good, so putting it back on. What could possibly be causing this? No check engine light since catalytic converter replaced. We are frustrated because nobody seems to be able to figure this out. Any suggestions? Sorry this is so long. Thanks.
**Also had plugs and wires done within five to six months.