Yeah, I think you'll like Zero Escape. I haven't played DC3R, but "multiple routes being key to the narrative" and "stayed up way too late after having my mind blown in the final route" instantly made me think ZE. The series has a lot of gameplay in the form of "escape the room" puzzles in between story sections. The characters aren't anything special, but the mysteries are gripping and the plot twists are spectacular. The soundtrack is great too.
And as has already been said, Steins;Gate is currently on sale in the US and is also really good. It doesn't reach the craziness of Zero Escape plot twists, but it makes up for that in characters and emotional highs (and lows). It's probably the only game that's made me cry multiple times T_T
Other stuff to check out:
Ever 17: We've just been discussing it above. It's the spiritual predecessor of the Zero Escape series - I think the big twist is better than ZE, but the quality is a bit inconsistent and I still prefer ZE overall. The first four routes are required for the true route.
Root Double: One of Ever 17's two main writers went on to make Zero Escape. Root Double is by the other guy. It never reaches the highs of E17 or ZE, but is still a solid sci-fi story overall. Has a pretty interesting route structure, following two different protagonists in wildly different environments before coming together for the final act. The choice system is pretty opaque, so you might want to use a guide.
To the Moon: Not a visual novel, but I think it's close enough to qualify and it's super cheap. Play this if you want to cry. No branching routes, but has some light puzzle gameplay.
VA-11 Hall-A: A small and personal story of a bartender in a cyberpunk dystopia. The characters and writing are really good, and the humour hit the spot for me. Serving different drinks gets you different dialogue, but there's no branching.
The House in Fata Morgana: VN-GAF's darling. It's really good, but doesn't fit your criteria quite as well. Heaps of intrigue and as far away from a high school setting as you can get, but isn't action or sci-fi and is a linear story with no gameplay. Stellar soundtrack and artwork, and a story that definitely stays with you long after you play it.