I will take a GUESS that the reason the drive won't work might be because the new MacBook Air isn't providing enough USB "bus power" through only only USB port.
Does the MacBook Air have only a single USBc port, or... does it have two?
Reason why I'm asking:
It might be possible to get a USB "Y-cable" and use that. With the Y-cable, the "single end" goes to the DVD/CD drive, and the two on the double end both get plugged into USB ports. This way, additional bus power is provided to get the drive running.
Did the DVD/CD drive come with a "power port" on the back?
If so, another solution might be to buy a power supply block and use it for power.
Make sure you research and get the right voltage, etc.
Final thought:
You could buy an external drive that IS NOT "a slimline" -- that is, it comes in a full-sized 5.25 case with it's own power supply.
Yes, it's "bigger"...
... But it should work without problems.