Is it possible to copy the remaining files of a (possibly corrupted) hard drive to an external hard drive?
Assuming you haven't already solved this, each of the following options should work.  Also not sure if you went through with a new computer:


Start up in the recovery partition of an external bootable drive (especially if you have a USB recovery installer of Mac OS - highly recommended). Use disk utility to restore your external hard drive from the internal drive of the old MacBook Air. Once completed you should have an exact clone to hook up to your new MacBook Air or investigate a fresh install of the old one.
This is a slight variation, but if you haven't already got a USB recovery installer, you could partition your external drive, make the first partition into the recovery installer, and use the second partition to restore into a clone.
Start in disk utility on your 2018 MacBook Air (or a friend's Mac). Use a Thunderbolt 2 to 3 adapter and start your 2013 MacBook Air in Target disk mode (hold T on startup). If your hard drive doesn't mount, hopefully it appears in the device list for you to access.
Dismantle the old MacBook Air and place the hard drive into an external enclosure. Hopefully you have a friend with an enclosure you can borrow. If you buy a full upgrade kit, you may be able to salvage the old MacBook Air as well!