Generally, the British common law doctrine of reception would apply here, even if not directly invoked (although most states used explicit language: e.g. chapter 2 of the West Virginia Code.) However, California is interesting because it did not do this. In ss. 20-23 of its Civil Code, it explicitly repealed everything before 1873 unless saved in the legislation. So theoretically California could go either way if it did split. That was then, however. I believe that it's fair to say that there were far fewer laws at the time, so "wiping the slate" was more feasible, but even back then there were criticisms of that approach. To do so now would essentially place the new state into a legal vacuum.
Although not an American example, something similar almost happened in Canada in the judicial reference case Re Manitoba Language Rights  1 SCR 721, 1985 CanLII 33 (SCC). It declared that, because the legislature had not enacted its laws in both English and French, as the Manitoba Act 1870 and Constitution Act 1867 said they had to, almost all the laws passed since 1890 were invalid. However, as this would produce "an invalid and ineffectual legal system" (question 2 and 3), the court granted a temporary extension of the validity of the invalid laws.
I think that this quote, from the paragraph after the previous quote, is instructive as to why a new state would not completely rewrite its laws:
The constitutional principle of the rule of law would be violated by
these consequences. The principle of rule of law, recognized in the
Constitution Acts of 1867 and 1982, has always been a fundamental
principle of the Canadian constitutional order. The rule of law
requires the creation and maintenance of an actual order of positive
laws to govern society. Law and order are indispensable elements of