Lots of the Otome Villainess genre these days has some element of court drama. Stuff like broken engagements, high-stakes tea parties, and spilled drinks on ball gowns are part and parcel with the genre, though they also tend to inject quite a bit of humor. Each of them have their own spin on things though, whether that be a unique magic system or what have you. Some recent ones I enjoyed would be I’ll Become a Villainess Who Goes Down in History, Villainess Level 99 (more on the humorous side), Endo and Kobayashi Live!, and Why Raeliana Ended Up at the Duke’s Mansion (a bit more serious).
You can check out the Royal Affairs tag on AniList to get a browseable list of royal-inclined anime, but it isn't going to be limited to Western-inspired courts since shows like The Apothecary Diaries or My Happy Marriage would also be listed there despite being set in China and Japan respectively.
A bit of a curveball is that a lot of the gundam or other space opera type series tend to have a large focus on political intrigue in addition to the battles. I remember that being a big theme when I watched Gundam Wing as well as Escaflowne and I have heard people say the same about Legend of the Galactic Heroes (but I haven't seen that one).