[obloquy]

Balrogs were originally Maia, creatures of Iluvatar. That is, they were created by Iluvatar. Melkor was a creature of Iluvatar, created by him. Melkor and the Balrogs were of a single nature: Valar.

I don't mean to be anal, but their "single nature" was Ainur, not Valar. The Valar (Melkor) were the higher beings among the Ainur, and the Maiar (Balrogs) were the servants of them.

The Dragons, by contrast, were of two natures. There are two possibilities: 1) Iluvatar created great serpents and created Maia, and Melkor corrupted them to evil and joined the Maia into the animal serpents. 2) Melkor made Dragons by turning normal serpents into Great serpents, investing them with Maia.

You seem to consider the Maiarin spirit as a necessity. It is not.

If the spirits of Melkor in Dragons were not Maia, what else could they be?

The seeming sentience of the Dragons could have been the original spirit of the beast that Morgoth corrupted, after having been taught to speak. Here is a quote that supports this as a possibility:

In summary: I think it must be assumed that 'talking' is not necessarily the sign of the possession of a 'rational soul' or fëa. The Orcs were beasts of humanized shape (to mock Men and Elves) deliberately perverted / converted into a more close resemblance to Men. Their 'talking' was really reeling off 'records' set in them by Melkor. Even their rebellious critical words — he knew about them. Melkor taught them speech and as they bred they inherited this; and they had just as much independence as have, say, dogs or horses of their human masters. This talking was largely echoic (cf. parrots), in The Lord of the Rings Sauron is said to have devised a language for them.
The same sort of thing may be said of Húan and the Eagles: they were taught language by the Valar, and raised to a higher level - but they still had no fëar.
But Finrod probably went too far in his assertion that Melkor could not wholly corrupt any work of Eru, or that Eru would (necessarily) interfere to abrogate the corruption, or to end the being of His own creatures because they had been corrupted and fallen into evil.

Myths Transformed

Granted, this essay was in regard to Orc origins, but it applies. It supports the idea that, in devising Dragons, Morgoth corrupted a serpent or reptile of some kind, and taught it to speak or to express his hatred and malice. It would also be capable of rebellion.

There's also the possibility that Dragons are the offspring of Balrogs and serpents. I mentioned this early on in the discussion. The result of such a union would not be more Maiar (cf. Melian + Thingol = Luthien), but rather a strain of perverted beasts with a measure of Maiarin blood, or possibly even sentience. This entire scenario could be the "corruption" that Morgoth imposed on the serpent in the above paragraph.