Man, I need to start solving my own problems, because at the expense of around 10 hours of my time and a little bit of sanity, I've finally fixed it! For any of you trying to make a movement FPS in Godot for whatever reason, you can use this as a base. It's klunky, but it works:
if (velocity.x <= 0 && -velocity.x <= max_speed || velocity.x <= 0 && velocity.x * direction.x < 0): velocity.x += direction.x * air_accel * delta;
if (velocity.z <= 0 && -velocity.z <= max_speed || velocity.z <= 0 && velocity.z * direction.z < 0): velocity.z += direction.z * air_accel * delta;
if (velocity.x >= 0 && velocity.x <= max_speed || velocity.x >= 0 && velocity.x * direction.x < 0): velocity.x += direction.x * air_accel * delta;
if (velocity.z >= 0 && velocity.z <= max_speed || velocity.z >= 0 && velocity.z * direction.z < 0): velocity.z += direction.z * air_accel * delta;
