Captain Saru canceled Yellow Alert on the bridge. Moments later when he was questioning Lieutenant Staments' ability to run the Spore Drive, you can see on all of the consoles that Yellow Alert has continued. First by a glowing yellow box on Ensign Tilly's console and then the two engineers at the Warp Core have icons stating Yellow Alert.
Dr. Culber is shown to be familiar with Trill symbionts even though in The Host (1991), set 110 years after Discovery disappeared, the existence of Trill symbionts was shown to be unknown to Starfleet doctors.
When Trill symbiotic creatures are introduced in The Next Generation (The Host) it was said that they were not able to exist long-term within humans. But no mention is made of how Adira has overcome this.
Burnham and Adira take shuttle 31 down to Trill. All of Discovery's shuttles were weaponized and deployed during the battle with Section 31 in Such Sweet Sorrow, Part 2 (2019). Also, there is no way to have fit 31 shuttles inside Discovery in the first place.
The computer's new voice has a distinct "breathy" quality to it. The computer has no need to breathe and therefore no reason to speak that way.
Michael asks whether there's any way of knowing if the Trill are as peaceful as they were a millennia ago. Millennia is plural, the singular form is millennium.
Adira complains that Burnham threw up in a wormhole, yet Burnham had told Adira that she had crash landed, then threw up, so Burnham was already out of the wormhole when she threw up.
Adira describes the Trill symbiont as a "squid" inside her. Trill symbionts in no way resemble squids. They are more slug-like than anything.