In the State of Massachusetts, the original state constitution in Part The Second, Chapter I, Section I, Article II says that:
Article II. No bill or resolve of the senate or house of representatives shall become a law, and have force as such, until it shall have been laid before the governor for his revisal; and if he, upon such revision, approve thereof, he shall signify his approbation by signing the same. But if he have any objection to the passing of such bill or resolve, he shall return the same, together with his objections thereto, in writing, to the senate or house of representatives, in whichsoever the same shall have originated; who shall enter the objections sent down by the governor, at large, on their records, and proceed to reconsider the said bill or resolve.
And from the New Hampshire constitution:
SEC. XLIV. Every bill which shall have passed both houses of the general court shall, before it becomes a law, be presented to the governor; if he approve, he shall sign it, but if not he shall return it, with his objections, to that house in which it shall have originated, who shall enter the objections at large on their journal, and proceed to reconsider it.
The Massachusetts state constitution was ratified in 1780 and the New Hampshire one did so in 1796, so it can be reasonably expanded that at least several states (Apparently not North Carolina) at that time did have executive veto power.