This is going to be annoying perhaps but I can't see a way to really address this without quoteblocks. I regret this.
Absolutely nobody is going to concern themselves with a refugee crisis when at any moment they could turn into an illithid and turn the refugee camp into a new illithid hive, and, importantly, they know exactly where they need to go to potentially prevent that eventuality.
You could make the argument that people would stay, especially with an arch-druid around, presicely because of that; kill the new illithid while they're weak and disorientated. There's no reason anyone would stand and gawk while the transformation happened either, and the new monsters would be killed mid 'birth' if the druids can't do anything.
Very little needs to be changed to fix this.
I disagree, I don't think anything needs to be changed. We've been told you have about 7 days to 'cure' the parasite (via the spell Restoration iirc) however talking to the fishermen if you save them from the trapped illithid mentions that it's days 3 or 4 by boat to Baldur's Gate, the only nearby large town which might have a healer capable of helping. If you consider your speed on foot you won't make it because by the 3rd-5th day you'll struggle to walk.
That's before the goblins are considered; the tiefling refugees are not from the local area, they were travelling and got attacked. They can't move on because of the goblins attacking local settlements such as the local village, and travellers. If there's enough to stop such a large group of people, that's a massive figurative and literal roadblock to anyone who needs to get to Baldur's Gate ASAP. This isn't even factoring in there are gnolls that are at least friendly to the goblins due to the Absolute.
Archdruids are (going off my memory) very powerful healers, so it's hardly a stretch to try and meet with the archdruid-which means dealing with the goblins.
The first is to remove any reference to anyone having seen Githyanki nearby, and to make the implication that nobody has ever seen gith before the crash. This ensures that while the ticking clock is stressful and threatening, there's no clear way to resolve it, and the player can decide how their character would go about trying to deal with a looming threat with hypothetical resolutions (greater restoration, gith shenanigans) but no clear path toward any of those solutions because they don't know where they would find a powerful enough healer, or make contact with the Githyanki.
I think the Githyanki are okay as they are; they managed to track a ship that hopped planes repeatedly, I don't think it's a stretch they tracked it down quickly when it crashed. That they're going around seemingly killing anything suggests they might not help, Laezel (your only interaction with them) is not exactly gregarious, and when they are finally met, that hope of being cured is briskly squashed and, barring specific dialogues, they try to kill you. That Laezel is also confused by this doesn't reassure.
I appreciate that they actually have a moment where characters say to you "What is going on, we should be suffering symptoms now" and Laezel panics at being ill before Halsin goes "The parasite is magically suspended"