Engine Stalls after replacing battery
New here, and I'm having a bit of an issue. I have a Mazda 3 2.0i w/ 85,000 miles.
Last Saturday, the car wouldn't turnover (partially due to the cold, partially due to it having a 5 year old battery). After a jump, I headed to Autozone, and put in their 'gold' level battery, and it starts like a charm. The bad news, it now stalls out when approaching stops. When my foot is not on the throttle, the RPM's go down to around 400-600 and bounce around when I driving under 30 mph or so. Sometimes they go up to 1,000 - 1,200, and sometimes it stalls out. I have no CEL (aside from when it stalls, when a whole bunch of lights pop up) and I was told by my local auto zone that they can't read what's going on if the CEL isn't on at the moment.
I brought it to my local dealership, got the 60,000 mile service and replaced a cracked motor mount. I told them about the issue, but they were unable to recreate it. I pointed them to the two TSB's (found through this site!) that greatly detail exactly how common this issue is and how to fix (PCM flash, which I was told was done with the 60k service and cleaning of the throttle body, which they said they checked). They also told me that after I unplugged the battery, I reset the cars computer, and it might take some time to relearn how the car has aged.
After two more days of continued stalling when slowing down, I brought it back to the dealership, and they recommended I get a fuel service where they clean the injectors. They AGAIN couldn't recreate the problem. I find this almost impossible to believe as I can't drive my car around the block with out noticing at least some odd RPM fluctuations.
So my questions are, how common is the problem of the fluctuating RPM's after changing a battery. Is this something that affects all cars? I can't imagine (for the life of me) someone at Mazda thinking it's a good idea to have a car act they way mine is acting after unhooking the battery. I'd think when weighing the pro's and con's with how our cars are built, the question "Does it have issues after unhooking the battery that might cause an accident" would be pretty high on the con list.
Also, what's my best bet?
Take it to another dealership: I'm not going back to the last one if they can't even recreate the problem, and I'm not 100% confident in their work. This is kind of funny because I was impressed that they found a cracked motor mount in my car. It led me to believe that they must have really looked over every aspect of my engine... until i logged onto this sight and saw that everyone and their mother seems to have problems with the motor mounts. Now I think they were scummy for charging me for replacing it and not applying my warranty (I have full coverage warranty up to 100k/2012 which I didn't know about until after).
A private mechanic: My concern here is they might not be familiar with the one-offs that affect only this model car
Be more patient: it's been a week since I changed the battery, started it up and drove 15-20 times, with multiple stalls both times
Bite the bullet and get the fuel injectors cleaned: this was the recomendation of the dealership, but from searching through this forum, I haven't seen a stalling issue fixed by this. I wouldn't be doing this myself as I'm not mechanically inclined, and it's going to be frigid the next few days.
I've searched around and read other people's stories. Most ended with people either cleaning the throttle body (which my dealership said was fine) or having the dealership flash the car's computer. While it's nice to see other people also have my problem, I haven't seen anyone recomend cleaning the fuel injectors. Could this be the issue? And if they need cleaning would a CEL be there?