Solution found! It's not the most elegant thing in the universe, but it's good enough until I get a better DAC.
At first, I was connecting my amp to the ODAC's 3.5 mm output with an RCA to 3.5 mm cable, and the noise was pretty bad.
The breakthrough came when I realized that the noise disappeared as soon as I touched the amp's ground plug with the ground of one of the RCA outputs on the DAC.
I then connected the amp to the DAC's RCA outputs with RCA to RCA cables, which decreased the noise, but not enough.
The final step was to solder a cable to the ground of one of the DAC's RCA sockets, then insert the other end of the cable into the amp's ground plug.
All noise is pretty much gone unless I turn up the amp's volume to ridiculous (unlistenable) levels and start doing certain things on the computer (e.g., stress the CPU, select and deselect multiple files with the mouse back and forth).
Below is my masterpiece