If you always end up in Washington DC and it looks the same every time, no matter what route you took, I think it's still a railroad. To the players, it might not feel like a railroad, which is a huge improvement.That misses the point of the term "railroad " entirely and broadens it to the point of being meaningless. It's a metaphor. A railroad is a singular linear path. A sandbox by definition can't be a railroad and even an AP isn't necessarily a railroad. A good AP is closer to a highway system: you always start in Boston and you always end up in Washington DC, but there are lots of different routs with different scenery to get there.
On the other hand, if the route you take really affects the end point, it's much less of a railroad. Take Red Hand of Doom, excepting the last chapter, which then would end up with a big fight. The big fight is genuinely affected depending on what actions was taken during the super-module. This makes Red Hand of Doom more of a sandbox than a railroad. Which made it a blast to run.
I recently ran an adventure, Reavers of Harkenworld, and it was written so that you could run it like a railroad, but at the same time, they did a great job describing the motivations/plans of the villains. This made it pretty easy to re-write the adventure from the PCs actions. For instance, my players ended up taking the keep first, then they staged an assassination on the Iron Circle leader and "escorted" the leaderless Iron Circle mercenaries out of the valley.
I was trying to prepare the session after they took the keep, but I had no idea of what the PCs were going to do, so I just prepared what kind of people the Iron Circle leader would have around him if he ended up fighting without his army. The players then decided to go after the army, scout it out. I had prepared for this too, giving the players some dice to decide the outcome of the fight. Since the PCs were not there, I gave small chances of a total victory for any side. It ended up with a stale mate, with the Iron Circle still not across the river.
The PC's started wanting description of the camp, because they wanted the Iron Circle leader dead. I then decided that the camp would be focused on the bridge/river, with the baggage train in the back an camp fires in front to light the river. The Rogue asked if he could make some slow-fuse fire bombs which he could plant in the baggage train, while the Paladin wanted to contact the villagers/elves to attack the Iron Circle if they wanted to get water from the river. This was all executed very well, they then bluffed/sneaked into the middle of the camp and got 2d6 rounds of uninterrupted boss-fighting time due to the chaos they had set up. It ended up with 4 rounds, which was just enough. At this point the PC's high-tailed it out, partially saved by the Rogue just slaughtering one of the lagging pursuers, spreading some more chaos.
This last bit was run completely sandbox-mode. The world was only put together as the PC's explored it. The preparation on my side coming down to thinking through some important anchor points: what the outcome of the battle would be, and what the NPCs would do if the PCs didn't act. They did act though and I could use some of that preparation to improvise the actual session. I don't think the players ever thought I was making it up at the spot.
It's a bit amusing how much Reavers of Harkenworld reminds me of a low-level Red Hand of Doom. Two of the best adventures I have run. Going to go with Madness at Gardmoore Abbey, which looks to be another adventure which can quite easily be adjusted to run more like a sandbox.
Anyway, when I try to run a sandbox, I never try to get the adventure or whatever I have written to have a particular outcome. I write up the motivations and only plan for likely actions/reactions of the NPCs. An AP is much more about how the PC's can react to the actions of the NPCs or pre-planned events. Sure, I can use an AP to create a sandbox, but that would very likely mean I would have to scrap a LOT of the content.