Should we notify our liability insurance about dissolution of our corporation and will/should it affect our coverage?

If the insurance was purchased in the name of the corporation, yes, you have to notify the insurance company, and dissolution of the corporation will terminate the policy;  it's possible that there might be a small refund of the premium, if the entire year was paid in advance.

From the question, I'm guessing that you want to continue the business, but not in the same corporation as now (since otherwise you wouldn't be concerned about the coverage).  You may be able to transfer the policy to the new entity you create to take over the business (another corporation, or an LLC, for example), but it's more likely that the new entity will have to take out its own new insurance policy.