"Сам/сама по себе" is a better translation of "on my/one's own" than "один/одна" (which is more accurately translated as "alone" in this context).
So assuming you didn't forget the leading "сам/сама" - report it.
Not really. "Я лежу сам по себе и думаю" should work, but by itself "сам" means "by myself" in the sense of "without any assistance". Given that lying is not the type of activity that is usually associated with needing help, your sentence sounds strange.