"battery is being charged at 12.9 V which is very"
Measure voltage at the alternator if 12.9v then suspect alternator.
Should hold 13.5v with all accessories/light on.

12.5 bat fully charged is probably best to replace it especially if 3-4 years old.

 

Yes voltage at the alternator is 12.9 V
Maybe it can be a loose serpentine belt

 

Engine shudder with ALL accessories on should not be much influenced by a bit low charging current. MOF, low charge may be caused simply by poor idle.
I'd start with checking tension on serpentine belt. 9 yr old vehicle chance is, it needs to be replaced.
Drop in RPM should be addressed, that's for sure. When was the last tune up done and what was done?

 

I’ve never done a tune up besides replacing engine oil, transmission oil,air filter and cabin are filters
What does a tune up consist of for the corolla ?

 

Hold idle speed to normal 650-700?
If still measures 12.9v with electrical load then suspect alternator if belt tight.
Belt tightness can be tested. Shouldn't be able to twist say more than 90 degress.

 

There are a bunch of posts on here about rough\low idle. The causes range from bad spark plugs, dirty MAF sensor, pcv valve, dirty throttle body, vacuum leak, to even a bad head gasket, which I just experienced. I would start with the easy stuff that should be done anyway and work your way up to the more involved items until your problem gets better.

 

Suggest you read for fault codes bet you find several.

 

If I had a dime for every report/question I've read about our generation Corolla 1.8 with the low idle/shake problem, because I was looking for them trying to find out what was causing mine, too.

I tried most everything mentioned short of changing my pcv valve, taking it to Toyota to reflash the computer, or check the compression for possible head gasket breach (which I'm not doing). I was hoping my recent alternator/belt change would fix it, but it didn't.

I've been doing most of my driving at night lately, and during the day using the AC, both which trigger a higher idle so haven't had to worry about it. If you don't find a solution and it's really bugging you then do what I thought of doing and cover your light sensor on the dash with some tape or something so that it thinks it's night time. The lights automatically will come on when you have the key on, and off when you turn it off.

 

Keep your headlights turned on and see if it helps, I cut my DRL wire and got in the habit on turning them on 24/7. IT helped fix my idle issue at stoplights or coming to a stop.

 

You can reset the idle by following these steps:

1. Start it up and drive it around for a while to warm it up. The car needs to be warmed up.
2. Turn off the car. Make sure all the accessories are off and unplugged. Turn off the radio, A/C, etc.
3. Turn the key on without starting it. Wait a couple seconds. Turn it back off. Do this twice.
4. Push the brake pedal. Put the emergency brake on, all the way.
5. Turn on the key, then start the car.
6. Put the car in drive.
7. Slowly let off the brake petal, but leave the E-brake on so you don't move forward. Just let the car idle. Don't press the gas. Don't get out of the car. Don't press the brakes (unless your car starts moving). Don't do anything. Just let it idle, in Drive, with the E-brake on for several minutes. Like 7 or 8 minutes just sit there. The computer will learn the new idle RPM.
8. Enjoy your new idle speed.