The modern Zeppelin uses two different lifting gases - hydrogen and hot air. No helium, and I'm sure you think that is kind of strange. Even airship captains ponder that choice from time to time, but it seems to have been the right one.
Helium is inert and a strong lifting agent, but its expensive and hard to come by, and frankly, you don't want to vent it if you can. Its also strategically important for certain industrial processes and super-cooling various computer components or medical scanners - the amount needed to lift an airship would cost more than all the other parts combined.
Hydrogen is well known to be flammable, and small - you can make tanks of solid steel or titanium, and the stuff will still slowly but surely leak out the crystal matrix of the metals. But that tiny size means more lift than helium, and you can manufacture it from natural gas or just plain water. Since no one is running helium collection operations at the moment, manufacturing hydrogen is a definite boon.
Hot air isn't as buoyant as the others, but you can manufacture it easily aboard the ship, and thus very actively control the ship in changing meteorological conditions.
Due to the strange and round about way the modern Zeppelin came about, more survived the event than heavier than air craft. Modern LTAs are based on an idea for space blimps - the atmosphere of most planets is either too thin, or non-existent for a normal drone aircraft to work, but a gasbag will float just fine. In turn, an airship is faster than a rover, but slower and closer to the surface than a satelite, some can even touch down for area studies. So yes, modern LTAs are built to standards needed to survive spaceflight.
It also helps that they're stored inside large metal shelters to protect the gasbags when not in use, rather than on the tarmac like planes. They weren't meant to be Faraday cages, but it was a happy coincidence.
Surviving the Event is different from surviving the aliens themselves. Its one thing to stage a transatlantic flight, or between two know safe areas. But you're not going to get to California by air - lasers, missiles, and alien warplanes are far too common to attempt that. Nor are these things cheap to run, at least relative to the resources available. But if you can afford it, a few tons can be transported, no problem. Pay up-front, of course.