Did people who lived before the law have the old sin nature?
Yes. From Adam to Moses, a period of about 2,500 years, sin or the Adamic nature was in the world. The pre-flood people had the old sin nature, and those between the flood and the giving of the law had a sin nature. They were no different than we are.
Les explains that verse 13 is a parenthesis that is hard to explain. He notes that 'until the law' refers to the period from Adam to Moses, about 2,500 years, during which sin or the Adamic nature was in the world. The pre-flood people had the old sin nature. The people between the flood and the giving of the law also had a sin nature; they were no different than we are. This connects to the earlier point that death passed upon all men and all have sinned because it started with Adam. Yet sin is not imputed when there is no law.
(For until the law sin was in the world: but sin is not imputed when there is no law.
More Questions
Why did God not impute sin before the law was given?
Before the law was given, God did not impute people's acts of disobedience because they were not per se breaking a law, even though they went against conscience.