Is a dative bond stuck next to a single covalent bond a double covalent bond?

If I understand your way, the nitrogen and oxygen of the dative bond will each have an upaired electron. If you want the O in the middle, you end up with two double bonds N=O=N. The doulb e bonds will be three e- from O and one from N. The reason the real structure is N=N=O has to do with "formal" charge which you may not have studied. In N=O=N the formal charge on O is +2 which is not preferred while in the other it is zero. The other rule of thumb not knowing anything is to put the LESS electronegative atom in the middle. That would be the N.