The second half of both of these doesn't work. "That" isn't suitable where you've put it – I think your getting confused with the construction "I was so [adjective] that [event]", meaning "[Event] happened because I was very [adjective]." So you might say "I was so tired that the accident happened abruptly" but, still, that doesn't really make sense because you being tired didn't cause the accident or cause it to be abrupt.
What you really mean is that you were asleep and the accident happened abruptly. I guess the point is that you didn't have time to wake up and figure out what was going on until the accident had already happened. But car crashes are (almost) always abrupt, so that part's redundant. Instead, you can just say something like
- I was catching some Z's/having a cat nap when the accident happened.
Now to address the question of whether it's idiomatic. I'd say that "catching some Z's" isn't idiomatic, especially when spoken to a police officer. I can just about imagine the most laid-back jazz musician saying something like that, but it would sound weird for me to say it. "Having a cat nap" is fine; I might not say it myself but that's just personal style. It's maybe more detail than is necessary, since the only relevant information is that you were asleep when the accident happened and it doesn't really matter how long you intended to sleep for. So I'd probably just say one of
- I was asleep when it happened.
- I was napping when it happened.
7Can I be so bold as to ask what brought this question on? – Mr Lister – 2017-01-03T12:17:28.107
6Just replace "that" with "when" and you're all set. But I'd recommend deleting "abruptly" as well, since it's hard to imagine a slow-happening accident in a moving vehicle. – Carl Witthoft – 2017-01-03T19:31:29.207
2@Carl Slow accidents do occur, but a sleeping passenger wouldn't know the difference, I imagine! – Kyle Strand – 2017-01-03T21:02:02.790
8"assistant seat" This is called the passenger seat, at least in American English. – jpmc26 – 2017-01-03T22:56:46.500
2Also, "your are sitting" may simply be a typo rather than a grammatical misunderstanding, but it should be "you are sitting in the passenger seat". "...is unconscious yet" is technically correct but archaic; the idiomatic phrase would be "is still unconscious." "Self-made" is understandable in context but not quite correct--it usually refers to something that made itself (e.g. a "self-made man"). "No one" usually refers to people, not objects, so your final sentence could more idiomatically start "If neither is natural..." – Kyle Strand – 2017-01-03T23:22:40.817
@MrLister to learn English while on the open road, and the tools for doing so...? – Ber – 2017-01-04T10:18:21.103