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.