I'm currently AirDroppping 3 very large files and I need to know if the files will continue to transfer while I'm not using one of the computers.
It should work as long as you have Wake for network access enabled in
Depending on the capabilities of your Mac, the option's description may differ:
The option "wake for Ethernet network access" is mentioned only to round up the answer. Wake on demand for ethernet will not help if your Macs communicate using Wi-Fi.
However, you can enable AirDrop to communicate using Ethernet as well as explained here. This will also enable AirDrop on so-called unsupported Macs.