If a contracting business shut down 7 years ago, can the business insurance carrier still be liable in a suit going back 10 years?

The statute of limitations varies from state to state and also depends on the type of case you have.  For example, there is a statute of limitations for personal injury, another for collection of a debt, etc.

A lawsuit can be filed in the state where the plaintiff resides or in the state where the defendant resides or in the state where the claim arose.  The statute of limitations may have expired in one state where a lawsuit can be filed, but possibly not in another where the lawsuit could be filed.

If you haven't filed the lawsuit yet and the applicable statute of limitations has expired, then your claim is barred and it is too late to file a lawsuit.

Most likely if you haven't filed the lawsuit and ten years have elapsed, the statute of limitations has probably expired, but without knowing the type of claim you have and the state(s) where it may be possible to file the lawsuit, it is not possible to give a more definitive answer.

If you had timely filed a lawsuit and the company went out of business, that alone does not invalidate the claims in your lawsuit.