Hi OP, I think you both have some work to do on conflict resolution
I would recommend you look at Jimmy On Relationships on Instagram and look into this as a good start. Every couple fights, and the longer term outcome is very much dependent on her bringing up something without criticism and you being very open to listening and resolving it from your end.
For example, when you say you are repeatedly apologising, are you going (in an exasperated way):
"Look, I said I was sorry, what else do you want me to do? I said I wasn't shouting, I was stressed, you just took it the wrong way!!"
Or are you going:
"I can see how upset you are and how my shouting was really difficult for you. I am really sorry that I did that. I don't want to make you feel scared or unable to tell me when I do something that upsets you. I was stressed and I reacted badly. I don't think I shouted, so maybe I'm not aware of how I come across. Can you help me understand so I can try and avoid it in the future?"
It is likely that the first leads to her being more upset because you are invalidating her feelings and telling her she is wrong, and the second will lead to a calmer discussion where you both feel heard.
I know it is SO DIFFICULT to apologise for something you didn't mean to do, but your relality is different from hers, and her feelings aren't wrong. Being defensive and only caring about how you are understood when someone else is hurt is extremely damaging..
If you are describing It honestly then it also sounds like she really needs to work on emotional regulation and bringing things up with you in a calm way that isn't overly critical. You can ask her, after she has calmed down, to talk about this. For example you can say:
"When you picked on my driving, whilst I was driving, it made me feel criticised and stressed. Driving is hard when you're being told to do something else! This may be why I wrongly shouted. Could you work on bringing things up in a calm way? This will help me respond in the best way for us to resolve it.
I think also it is really important to consider how sometimes the arguement is worse than the actual thing. This is how it is for me. My partner may do a small thing, and when he repeatedly denies, invalidates or victim blames me it quickly goes wrong and I end up very upset. If he took the approach above of validating my feelings and trying to understand his actions, instead of telling me I'm wrong for being upset, things would be a million times better.
Good luck op