Big question is Do You Have a Backup?
I had a very similar problem installing High Sierra on a mid-2012 Macbook Pro. Unfortunately Apple doesn't tell you that this upgrade is pretty major as you are changing file systems. There are many people experiencing this or very similar problems.
I did a ton of research, none of it very helpful. Running first aid over and over did nothing, and you can't reinstall Sierra because of the file system change made by the High Sierra install.
My solution was to get the machine back to a pristine state with Sierra installed as though you just got it and then do the upgrade to High Sierra via the App store.
Here's what I ended up doing, YMMV:
Make a bootable Sierra (NOT High Sierra) thumbdrive (How to create a bootable installer for macOS - Apple Support)
You will need access to another Mac to do this.
Boot the target machine and run disk utilities.
Reformat the hard drive (this is why you need a back-up - any data will be lost) use the default settings. In my case I had to reformat the drive and run first aid on the physical drive and volumes several times before it all looked right in Disk Utilities. In my case I could not get the mechanical drive to format properly and replaced it with an SSD (probably can't do that with an Air).
Install Sierra.
Boot the machine and go to the App store and you should now be able to upgrade to High Sierra.
If you have Time Machine, migrate the data onto your machine (I am told this works flawlessly despite the fact the Time Machine file was created under Sierra. I did not do this myself, my situation was far worse! I had to recover using a corrupt Crashplan backup. Nightmare). If you have only a regular backup, you will want to restore the data files, and then, pick and choose what you can salvage from the user library / applications / documents directories. You may lose calendar, contacts, etc., I did. Restoring some of those files can cause apps to crash because of indexing issues, etc.
Hope this is of some help. Good luck.