The Skaal are descended from the Nords who served the Dragon Cult on Solstheim, and built tombs to honor the Dragon Priests who ruled over them. 

The Guardian confronted the Traitor, and the two fought a terrible battle. So great was their power, Solstheim was sundered from Skyrim.

It's been millennia since the Dragon Cults were around, which means the Skaal's culture/language would have evolved a lot. 

The current Nords of Skyrim as well as everyone else on Tamriel at least speak Cyrodiilic, but the Skaal are a highly secluded, small tribe. Secluded cultures ALWAYS evolve differently from other cultures. Since the Skaal do interact with travelers, they are able to communicate with them (that is, at least their leaders speak Cyrodiilic). They would also very likely have Cyrodiilic loan words. But their primary language, I think, would be a similar but different language from Cyrodiilic. Basically, the two languages would be dialects, daughter languages of Dovahzul and Nedic that branched off during the ancient Nordic language period (the period before the Cyrodiilic Empire formed and the Common Language was formed/spread).

Does this make sense?

Obviously they have similar language roots, judging by their villager names (bolded parts share roots/names with mainland Nord names; italicized names/parts are not Nord or Scandinavian):

Bera, Baldor, Deor, Yrsa, Edla (an Oblivion character), Nikulas, Fanari, Tharstan (Tharsten), Morwen ("wen" unusual for a Nord - elvish suffix), Aeta, Finna, Oslaf, Frea, Storn, Wulf

Elvish influence in their names? None of them are Dov-related. Perhaps it could be argued that Scandinavian names are an aspect of Nedic culture, and that's why the Skaal and mainland Nords share name styles. I still think they would speak a variant of "Norren" (aka the Dovahzul-influenced Nedic language, basically Old Norse with a twist).

(Yes, I know that in the game they are "just Nord", but I'm thinking deeper, here!)

I think the theory that they'd have their own language works but perhaps only up to a certain point in history (say, pre-settlement of Solstheim by the Empire). Judging from what we see of them in Dragonborn some 200 years later, they appear to have fully adopted the Cyrodiilic language and there isn't much sign of the language they may have once spoken. They are certainly culturally distinct from the "mainland" Nords, and I would argue their culture is much closer to their Atmoran ancestors than Skyrim's Nords are.

UESP mentions "Like on the mainland, the Nords shared the island with the native elves." If they interacted with these Snow Elves enough, this could explain elvish influence on their names.

I always like to compare Dovahzul to Latin as it related to the Nords. This is all speculation but in the same way the Catholic Church used Latin for mass, the Dragon Priests (probably) would have used Dovahzul to perform rituals or whatever else Dragon Priests did. What's interesting to your question is that in real history, Latin as used by the church developed into its own "style" called Ecclesiastical Latin. It's conceivable the Skaal (or Nords in general) would have had an "Ecclesiastical" Dovahzul.

Another big question is what happens to Dovahzul after the Dragon War and the fall of the Dragon Cult? In mainland Skyrim I don't see Dovahzul surviving past the Dragon Cult except through the Nordic Tongues. Maybe it happens differently with the Skaal.

Well, the game makes everyone speak Cyrodiilic except for some expletives from Dunmer because if they didn't, it would just frustrate people.

Anyway, I still think, even if Dovahzul was the language of ritual, some words would have seeped into mainstream culture and likely stayed there, especially if those words were basic, mundane things said often. And languages don't just up and die because an oppressive regime goes away.