I think (but am not certain) that you're treating "sandbox" and "railroad" as two ends of a spectrum. If you are, I think that can be too limiting, and make it harder to see a way out of your problem.
I'll try to explain - to the extent that the explanation is redundant or obvious, I apologise.
I'll start with an example from my own game. The PCs were busting up a lair of goblin slavers (The Chamber of Eyes in H2). In the lair were some duergar slave traders. Per the module, the duergar attack when the PCs enter their room. I ignored that particular bit of GM advice - when the PCs entered the duergar's room to take a breather (short rest) from fighting goblins, I started with a standoff and then had the duergar enter tentative negotiations with the PCs. Resolved as a skill challenge, it ended up that the PCs first learned that the duergar had purchased slaves from the goblins, and then agreed to ransom those slaves for a certain amount of gp from a neutral city in a month's time.
Generalising a bit from the example: the adventure "came to the PCs", insofar as they encountered the duergar during a mission, but the resolution was driven by the players, and at the start of the encounter neither I nor them was expecting the upshot to be a ransom contract. And the PCs also then got a reason to travel to a new part of the campaign world (the neutral city).
If you present your players with these sorts of encounters, where the options for resolution are open-ended, but the ingame stakes (rescuing prisoners, averting war, stopping the cult, etc) are high enough that the players will engage, what happens? Do they roleplay their PCs and make choices and bargain for things? Or do they sit there waiting for you to feed them the script?
If the latter, then maybe there's no hope for this particular group. But if the former, then is that a way that you can drive them to engage the fiction a bit more, and see a bit more of the gameworld, but without them having to initiate the adventure by talking to non-antagonistic NPCs, picking up rumours, etc. (Which is what I see as the core of the sandbox.)