Check engine light, engine misfire

I have a 2003 Saturn Ion. About 3 weeks ago it started sputtering while idling at stop signs with the check engine light on. Then the next day it ran great to wherever we drove it, but upon starting it up it would have the same problem and while driving it home it had no pickup and would drive very slow and the check engine light would flash as well.

We took it to Firestone Auto Care who did a diagnostic which came up with codes for the oxygen censor and catalytic converter. They replaced the oxygen sensor after which it ran fine with no more codes. Then about a week later the problem came back.

We took it back to Firestone who did a second diagnostic which came up a catalytic code. They recommended we take it to Midas since Firestone does not do welding. Midas replaced the converter that they determined needed to be replaced because small pieces of the inside had broken off and were loose.

The car ran fine up til the day before yesterday when it ran fine to a place, sat parked and was started. The car sputtered had no pickup and the check engine light flashed I as drove it home. I took it back to Midas who determined that problem might be a blockage of the exhaust system due to broken catalytic converter pieces. So they dropped the exhaust system and checked for loose pieces and sure enough there were piece of the converter in the resonator and muffler. They tried to clean the old parts as best they could but could not remove all the pieces which could cause blockage. So we had them replace the resonator and muffler. After they rebuilt the exhaust system it ran for a while, the check engine light came on for them. It started sputtering and carrying on like it did before any of the work was done. They said there was no diagnostic code but that there was a definite misfire of the engine. They gave up had me pick up my car now with new parts at no cost. Now I'm not sure what should be done or where to take it.