I absolutely love the map you provided and it was a nice touch to include cropped images of each portion next to their descriptions. The corner images on each page are beautiful and I thought you picked a great image for the cover.

I liked the intricate relationships, backstory, and the character descriptions were all great. A nice addition would be stat blocks, to save the GM having to create their own.

You presented an interesting story, but I think the introduction could be simplified, there a lot of names and concepts for readers to keep track of:

Goldgreaves (dwarves, graybeards, dwarven soldiers), Princess Ciswin,  red wizards (mages), Scarlet Spires (halls and tower, living quarters and fortifications), General Gripfast, Greybeard Graspfast, old King's Quarters, Throne Room, the most dangerous of spells (dungeon dimension, demons, convoluted ritual, giant betentacled monsters), the valley(?), headmaster Kanae, Death Wizard sentries, Marlow (school's ambassador).

Instead of multiple paragraphs maybe each topic in the introduction could have a separate bullet or heading for quick reference.

I like the dungeon design, its very usable. The flooded grate creating a waterfall across three levels makes it feel connected and alive.

If I ran this I would try and come up with a way to actively engage the players into the throne room scene, it would take some serious tweaking, but possibly placing them in the throne room after the dungeon delve - explaining how it played out.

The Dungeons In Between has a dope three dimensionality  that's hard to convey in a static image. This makes it easier to wrap your head around as an interconnected space. Splitting the salient portions up on the subsequent pages to illustrate the exact position of the party is quite effective.