The officers and the directors have the fiduciary duty to the corporation and the shareholders. While the owners (shareholders) are shielded by the limited liability, the directors and the officers (who in case of a small business are frequently the owners themselves) are not, and the State can go after them for all the fees and penalties they failed to pay on behalf of the corporation.
So if you own your own corporation of which you're the director and the CEO - the State can go after you, not because you're the owner, but because you're the director/officer.
Also, a corporation that is not current on its regulatory obligations may lose its corporate veil, making the owners vulnerable for the liability arising from the corporation. In which case, the debtors (including the State) can go after the owners themselves.
I suggest talking to a lawyer licensed in DE about how to handle the situation.