It's been years since I've seen TAS. Do they say anything to indicate that it's new? Like, "Hey, Mr. Spock, check out the new turbolift!"
No, although TAS certainly acknowledges that the animated episodes are happening after TOS. Several episodes, of course, are direct sequels to TOS episodes ("More Tribbles, More Troubles", "Yesteryear", "Once Upon a Planet" and "Mudd's Passion", or have a returning villain - ie. Kor in "The Time Trap").
Also, Arex and M'Ress, about to lunch together in "The Practical Joker", get blamed as the pranksters and the inference seems to be because they are both new to the ship they are under suspicion.
No one mentions that the bridge now has a second turbolift (although it still does in TMP) nor that the bridge chairs now have round bases on them.
Roddenberry's take on stardates in TOS (and continued by DC Fontana in TAS) was that they are quite random, due to position in time and space, but that
within an episode the numbers are shown to move forward with the passage of time. TAS stardates are quite random, episode by episode.
The very useful timeline that appears in the recent ST novel guidebook, "Voyages of Imagination" by Jeff Ayers, uses production order for TOS, and "Star Trek Log" publication order for TAS, which makes use of the revised (sequentially later) stardates added by Alan Dean Foster when he adapted and serialized them for book form.