I moved to California and while in California I was sued. I moved back to NY. The statute of limitations should protect me?

You have to give some explanation about the situation.

Even on the New York State Bar Examination, aspiring lawyers get a factual 'pattern' upon which to apply a rule or principle. Here, you provide us no facts except that you ran from a law suit in California. Based upon that, I would conjecture that your move does nothing to protect you from litigation as all states have tolling and extending statutes that allow a law suit to go on ice until the defendant can be located and served. Here I cannot answer anything else because I have no idea what the operative facts of your question are.

The statute of limitations applies to the party suing you. If, for instance, they sued you for a car accident in California which has a two year statute of limitations, they must file suit within those two years no matter where you've moved to now, and no matter what the statute of limitations is in the state you've moved to.

Where were you served with process? In California or New York? What is the underlying cause of action and date of accrual. More facts are needed to respond to this question.