

Yes, they did - on the face of it - switch to a Berman-era Stardate system of 1000 stardates to a year and 2364 (TNG Season 1) as a baseline for 41000.
But things got weird really fast.
So Burnham travels 930 years ahead of Discovery’s original time (2258) to land in 3188 (as her suit computer said in DIS: “That Hope is You, Part 1”) and spends a year in that time before Discovery itself showed up (DIS: “People of Earth”), so the year should have advanced to 3189.
Yet Burnham’s log in “People of Earth” detailing how she spent her year without Discovery is dated 865211.3. If we work backwards to TNG: “Encounter at Farpoint”, which took place on Stardate 41153.7 in the year 2364 (TNG: “The Neutral Zone”), that makes the 865000s the year 3188 instead, which can’t really be.
Furthermore, if we are following the 1000 stardates equals 1 year convention, 221 stardate units brings us only to March 22 of that year, so we can’t even say that she landed at the start of 3188 and made her log at the end of that year.
Then comes Season 4’s DIS: “All is Possible”, which has the stardate 865661.2, allegedly a week after the previous episode DIS: “Choose to Live”, which places it back in 3188, and 661 stardate units takes us only to around August 29!
Season 4 starts five months after the end of Season 3 (as stated in Season 4’s premiere DIS: “Kobayashi Maru”), and it’s highly unlikely that Season 3 took place in the space of one month (between March and August is only six months). So whichever way you slice it, the Stardates are off by at least a year, if we’re still following the TNG convention.
But that’s not all. Season 5’s DIS: “Under the Twin Moons” is on Stardate 866274.3, which places it in 3189. However, this is also an impossibility since, as I’ve noted, Burnham arrived in the 32nd Century in 3188, then spent a year before reuniting with Discovery (3189), then months passed between Seasons 3 and 4, and also between Seasons 4 and 5, so at a minimum it should be 3190. And in the very next episode DIS: “Jinaal”, they definitively call the year as 3191.
If we lived in a sane universe, DIS Season 3 would have covered Stardates 865000-866999 (3188-3189), Season 4 Stardates 867000-867999 (3190), and Season 5 Stardates 868000 onwards (3191) and we could breathe a sigh of relief. But the given stardates don’t.
One way to resolve it is to throw out what we knew about TNG stardates and just live with the idea that the 1000 stardate units stretch out over the course of 2-3 years. However, that idea makes this old Trek chronologist’s face twitch.

SFA: “Vitus Reflux” starts with Darem saying it’s Stardate 868858.7, which works out to 3191, so that was very satisfying to hear.