In my experience, there is always a "clumsy" period in learning a skill which involves maintaining multiple concurrent physical processes. I don't believe working memory has much to do with overcoming it. It's more typically a case of "wrapping" a complex action into one you can carry out on demand. Right now, pulling over seems like it involves a lot of minor actions (mirror, signal, break, mirror, clutch, gear, cancel signal, handbrake...), but your brain eventually wraps it all up into a single major action.
I would advise changing your instructor if they're a source of stress. Even as a skilled driver, a stressful passenger can seriously impair your ability to concentrate.
I've been trying to learn how to drive and unfortunately I suck at it. Some combination of a stressful teacher and hyperfocusing have made it very difficult to learn. My biggest problem is with the multitasking aspect. Remembering to put on the turn signal while stopping and and checking my speed and watching out for other cars, etc. It's difficult for me, I forget or miss things. One thing I was considering may possibly help is using dual n-back to boost my working memory. Does anyone have any thoughts on the likely effectiveness of this?