Oooooooooooh I love Taiga drama!

The broad facts are correct, like Masamune's close confidants, how he came to lead the Date clan and the fate of his parents, the changes in politics and territory of the Date clan, and that Masamune decided to support Tokugawa Ieyasu's side after Hideyoshi's death.

The details on the other hand are a bit iffy. On the one hand, the drama, as expected of Taiga Dramas, actually read/paraphrased to the audience Masamune's letter to his mother complaining about the Korean campaign, which is fantastic. And the drama gave a very balanced, much more plausible depiction of Masamune lying his way out of being executed by Hideyoshi than is depicted in the narrative sources. On the other, while death of Masamune's brother and the exiling of his mother, as well as the exile of his top general Date Shigezane and the death of Shigezane's wife, are kind of shrouded in mystery, some evidence exists (like the timing and location of the characters that we can put together from correspondences) that it probably didn't happen the way the drama depicted. And, again as expected from Taiga Drama, the drama used things that are recorded as rumours from not very reliable sources that historians don't really believe happened and depicted them as facts, like Masamune trying to poison Gamo Ujisato or Masamune carrying a golden cross into Kyōto to ask for Hideyoshi's forgiveness (admittedly, the drama treats the former incident as misunderstanding from Ujisato).

However, one thing that the drama, once again as expected of Taiga drama, did do a lot of twisting of the narrative was to play up the role the main character, in this case Date Masamune, and his clan. For example:

1. The drama implies that Shigezane's heroic flanking attack at Hitotoribashi turned the loosing battle into a heroic victory by failing to mention that all Shigezane's attack do was buy Masamune enough time to escape with his life (admittedly still very damned important). The battle was an overwhelming, though expected, loss for the Date clan. The anti-Date army retreated after the battle not because they lost, but because in their absence the Hōjō and Satomi had invaded their lands.
2. The drama depicted Masamune as constantly trying to win overlordship of Japan, even after Sekigahara and the establishment of the Edo Bakufu, but there's not really any evidence for this except rumors of it in Italy (of all places) from Hasekura Tsunenaga's expedition. While it doesn't go against the facts, it does kind of force the interpretation, as from what could be interpreted is that Masamune was a loyal Tokugawa ally from prior to Sekigahara.
3. The drama turns turmoil in Masamune's circle during Hideyoshi's years into personal disagreement with an overly conservative Shigezane by failing to mention that Shigeniwa Tsunamoto also went into temporary exile, and Shigezane wasn't the only one.
4. In the drama, Ieyasu on his death-bed informally named Masamune "deputy-Shōgun" and while no one else was around to hear it Masamune started referring to himself as such on occasion. There's no record of Ieyasu doing such things (Taiga dramas seem to like doing this with whichever main character who's going to outlive Ieyasu), and while it's true that Masamune continued to play an important part in Bakufu politics, he never gained enough power in the Bakufu to be even remotely considered such (also the Bakufu didn't appoint official deputy Shōguns, but to be fair the drama implies it's just something Masamune styles himself).
5. The Bakufu must have known very well the Date clan was cheating the Bakufu's law of one-castle-per-domain, but there's still nothing to indicate that it was going to use this as a pretext to militarily crush the Date clan. After all, the Date clan was one of the Bakufu's strongest supporters among the Tozama, and also had one of the closest ties.

Also if we want to be really picky, the character would've been Idate Masamune and his clan the Idate clan, as the name was then pronounced, not Date.