Danes spoke Danish. At least they themselves would have referred to it as such, it's not as if there's any point in the past 1000 years where they suddenly they felt they were speaking a different language from what any other living Danes were.

The problem here is that "Old Norse" is a rather vague term. Most "Old Norse" you see is the language of the Sagas and Eddas written down in Iceland in the 1100s-1300s. But also the language of contemporary Norwegians as in the book Stjórn, which by that point was only subtly different from the language spoken in Iceland.

Old Norse in the broader sense encompasses all the Norse/North-Germanic languages from about 800-1300. These were mutually intelligible. The bulk of the changes that set modern Danish (and Swedish and Norwegian) apart from Old Norse occurred in the 1300s, when you have a massive influx of German loan words on one hand, you have a loss of much of the grammatical case system (although that began earlier with Danish than Swedish/Norwegian). In the case of Danish the weakening of final vowels into a schwa sound written 'e', the 'softening' of a number of consonants after vowels (t to d) and the development of stød. These latter things also set Danish apart from the East-Norse Swedish. (so by the late middle ages there are a number of commonalities between Norwegian and Swedish not shared with Danish, despite Swedish and Danish being East Norse and closer at the end of the Viking Age)

Dialectal variations had always existed. The split into East and West Norse is not a split into two mutually unintelligible languages, but rather a grouping of dialects based on a particular set of changes that they had in common. For instance, in East Norse words like jarn and hjarta became jærn and hjærta while remaining the same in West Norse. West Norse ei as in steinn, bein became e so sten, ben in East Norse.

The 'standard' Old Norse is what you could call more specifically 'normalized late old West Norse'. 'West' because it's from Norway/Iceland, 'late' as it is in fact from around the 1200s and there are some subtle differences to Viking Age West Norse, and 'normalized' meaning it has a standardized spelling which is based off modern Icelandic spelling. In fact it's constructed to make the written Old Norse more understandable for modern Icelanders; for instance, Old Norse did not distinguish long and short vowels: an 'a' was an 'a' (and in fact the difference in vowel length wasn't so substantial, given things like texts rhyming var with sár). Normalized script uses accents, writing 'á' for a long 'a'. This is in part because the long-a sound developed into an entirely different sound in modern Icelandic and this is masked a bit by using 'á' for both. So they appear much more similar in writing than spoken. (Old Norse á is like in father but Icelandic á is like ouch)

The reason for this state of affairs, is that despite the fact that a significant majority of Scandinavians spoke East Norse dialects, it was the Norwegians and Icelanders (especially the Icelanders on a per-capita basis) who did the most writing in the vernacular language. Official letters and documents in Sweden and Denmark in the 1100s and 1200s were mostly in Latin. It was mainly the Icelanders who wrote down stuff that posterity would be most interested in - namely their own folklore of the era and earlier.

Nevertheless Old East Norse dialects are by far the more attested ones from the actual Viking Age, as the vast majority of runic inscriptions from that era are from Sweden (thousands), secondarily Denmark (hundreds), but fewer than 100 from Iceland and Norway together.

Anyway. The short answer is that the period given in Denmark to the Danish language (by NÅ Nielsen) from 800-1100 is usually Olddansk ('Ancient Danish') or Runedansk ('Runic Danish'), followed by 1100-1350 or so being Gammeldansk ('Old Danish') or Ældere Middeldansk ('Older Middle Danish', by Katlev in Politikens Etymologisk Ordbog, 2000)

But these are all just sub-groupings and periodizations of what is still referred to as Old Norse in a broad context. Nevertheless, almost all "Old Norse" dictionaries you will find will be for Normalized West Norse, and the same goes for most materials on Old Norse, particularly those available in English. No English-East Norse dictionary exists (yet). It would also be hard to write one since there's also no 'normalized' form of Old East Norse with standard spellings as there is for West Norse; you're left with the spellings people actually used, which were often inconsistent. (as said, long and short vowels were not distinguished, nor was þ/ð or v/u etc)

So the only route that's really practicable to learn Viking Age Danish is to learn 13th century Old Icelandic, and additionally learn about the dialectal differences and how things changed over time. Those things are fairly regular and very minor compared to learning the whole language and grammar. Even in academic contexts like Samnordisk Rundatabas (the common Nordic database of runic inscriptions) they have a field for translations to Normalized West Norse, even though most texts are East Norse as said. But it's practical since if you want to search for a particular word it's much easier to find it when there's a single spelling to look for.

In practice there aren't any people who've studied OEN without also studying OWN, although there are definitely some who've studied OWN without OEN.