Every language has lots of varieties which differ in conservativity among themselves. This effect can be massive. If any particular methodology for assessing conservativity forces a choice between the spoken form and the written form, or between various available registers early in the process, such a decision may drastically influence the outcome.
Conservativity is a multidimensional quantity. Any of pronunciation, writing system, vocabulary, morphology, syntax, semantics, pragmatics could be used as a basis for comparison with potentially different results. Looking exclusively at the pronunciation or the writing system obviously excludes some key language varieties from the comparison; and semantics and pragmatics are difficult to control for exo-linguistic factors of societal change. Let me therefore try and look at the basic vocabulary of the languages involved.
When several languages have a common ancestor, lexical differences between each one and that ancestor can be attributed to a combination of vertical (inner) influences and horizontal (external) influences. Vertical forces include sound laws affecting the form of words and meaning shifts affecting the semantic fields: a word can "wear out" and have to be replaced in its primary meaning under the pressure of acquiring secondary meanings. Horizontal forces include borrowings from other, not necessarily similar or genealogically related, languages.
Lundin has, in his comment on the OP, very aptly characterized a completely general problem which surfaces whenever a native or equally fluent speaker of one of the languages being compared looks at the other language, lesser known to them. ("Some words in modern Norwegian sound like old fashioned Swedish to Swedes and vice versa.") Vertical lexical differences tend to suggest that the lesser known language is the more archaic one, while the objective situation can be perfectly symmetric or even opposite to the apparent one. This happens when the common ancestor language offers several near-synonyms (words with overlapping semantic fields) and the descendant languages sort out the overlap differently; educated users of either language are likely to be familiar with reflections of both words in both languages for some time, using "their own" lexical item actively and recognizing "the other" item passively as indicative of either a non-standard dialect or archaic usage. The item is then subconsciously interpreted as a marker of "archaic" or "conservative" language use even in the other language, and this feeling might persist subconsciously maybe even after the speaker is made aware that the usage is entirely common and unmarked in that other language. This "subjectivity" of comparison is perhaps less of a problem with horizontal forces inside the traditionally linguistically sensitive Europe, as long as the external origin of loanwords to either language remains equally obvious to both its respective native speakers and to its foreigners.
To avoid drowning in the problem of subjectivity, it is practical to use a standardized battery of "core meanings" for the multi-way comparison (between the ancestor language and each descendant language). Constructing such a battery is a tricky business because we need to select concepts which tend to exist in every language in the world and have a single unmarked word to denote them. This task has been tackled by glottochronologists, whose basic tools were the Swadesh's lists.
Let me try quite unscientifically comparing Swedish against Danish and then also Norwegian (bokmål) using the first 100 words from the list.
- 44 identical items. (I did my best to land the items here when either language offered multiple lexical items or forms, per my bias toward a drawn result.)
- 26 times the Danish word showed a vowel reduction compared to the more conservative Swedish spelling. Bokmål sides with Danish on these.
- 5 relative pronouns (or question words) preserve a more conservative spelling with initial "hv-" in Danish compared to Swedish "v-"; however, the "h" is silent in Danish with no spoken difference. Norwegian is more divergent here, often reducing the cluster to /k/, which could be interpreted as being more conservative than either Swedish or Danish - while Danish is the most conservative one graphically on this overrepresented part of the lexicon.
- 8 items look as if there's some German spelling influence over Danish, often spelling "nd" where Old Norse and most descendants had just a "n"; however, the "d" is silent in Danish, unlike in German. Norwegian tends to be written like Swedish here.
- 7 items where Danish and Swedish preserved different lexemes from Old Norse. Norwegian tends to preserve both lexemes side by side, or dependent on the dialect. (I will count each one as a draw.)
- 1 more such item (Danish "ikke" for "not") where it seems fair to say that Danish (and Icelandic) are the innovators here.
- 3 items where Danish reflects an older "g" as "v", unlike the rest of the flock.
- 1 item ("två") where Swedish preserves a "v" lost in Danish.
- 1 item ("träd", meaning "a tree") where the Swedish word contains a fossilized definite article (which is so well baked in that it doesn't stand in the way of forming "trädet", meaning "the tree", yet another definite article now that the old one had been lexicalized)
- 2 items (Danish "hånd" vs. Swedish "hand"; and Danish "lille" vs. Swedish "liten") where I was unable to make up my mind about who is being more conservative here.
- 2 more items I lost track of (sorry for that), one where the Danish form superficially resembled Icelandic more than the Swedish form, and the other was vice versa; but I failed to check Old Norse for these, so this is in itself quite inconclusive and I will count these as two draws.
(I haven't specifically elaborated on too many Norwegian items, and that's mainly because bokmål appears (judging only from these top 100 lexical items) to have been so much influenced by Danish, Swedish, or both, that the complicated question of conservativity of Norwegian could perhaps be approximated by a simpler question of tracking down those influences on the written and spoken form of this or that dialect and register of Norwegian.)
In retrospective interpretation of several of my findings, I have greatly benefitted from this resource and this resource, especially in cases where it was not obvious which form had been the original one.
Time to score - written forms:
- Swedish emerges as more conservative than Danish: 26 + 8 + 1 + 3 + 1 = 39
- Danish emerges as more conservative than Swedish: 5 + 1 = 6
- Draw: 44 + 7 + 2 + 2 = 55
Time to score - spoken forms:
- Swedish emerges as more conservative than Danish: 26 + 1 + 3 + 1 = 31
- Danish emerges as more conservative than Swedish: 1
- Draw: 44 + 5 + 8 + 7 + 2 + 2 = 68
Don't take my totals too seriously as I don't speak those languages myself and I was sure to miss many important points especially in the vocalic space, and I was biased toward a drawn match.
However, if either language has turned out to be more conservative than the other, it has been Swedish, and it has been so mostly because of extensive vocalic reduction in unstressed syllables, as well as non-historic "-nd-" and "-ld-" clusters found in written Danish. I may conjecture that some of this is a result of German historically expending a little bit more of an influence over the evolution of Danish compared to the undetected influence of, for example, Uralic languages over Swedish. But it would be a mistake to consider this explanation to be the only possible one, or to consider the overall effect strong.
Remember that a close look at morphology or even at the lexicon as a whole could hypothetically arrive at the opposite conclusion; the method above is unable to detect whether a language tends to solve its new naming needs through derivation from old morphemes or through borrowing, which is something where Icelandic draws a lot of its fame for being conservative within its subfamily from. (It's rare to encounter an internationalism in Icelandic.)