> The conversion from 16 bits to 7 bits was using rounding.

As a very quick fix, you can dither by just adding a random value from [-0.5, +0.5] before rounding (to -64..+63 or whatever your range is). It will give you a dither, and probably sound slightly better; a bit more noise for much less distortion. Noise shaping is left as an exercise for the reader :-) (It is probably nontrivial to get perfect with variable sample rate anyway.)

> I used a custom NES emulator to emulate the generated inputs, and I had it count CPU cycles so I can convert that into seconds, then parse the .wav file with that info.

It sounds like you are just picking one sample without any filtering/averaging/anything (nearest neighbor); this will cause aliasing, which is another part of the reason for the “roughness” you may hear in the sound. You can do a very cheap trick here as well: Take some audio software you trust (say, Audacity) and convert the .wav file to 25208 Hz. This means that you'll get good filtering for most of your audio, and less bad filtering for the 13.85kHz parts.