Are you sure the pins are correct?

Maybe use a smaller random number generator per https://arduino.stackexchange.com/a/18092/6628 ?

static uint8_t lfsr = 0x01;
const byte LFSR_MASK = 0x8e;

void setup() {
  pinMode(0, OUTPUT);
  pinMode(1, OUTPUT);
  pinMode(3, INPUT);
}
void loop() {
  //      int buttonState = digitalRead(3);
  if (digitalRead(3)) {
    if (generateNoise()) {
      digitalWrite(0, HIGH);
    }
    else {
      digitalWrite(1, HIGH);
    }
    for (int i = 31; i; i--)_delay_us(64516);
    digitalWrite(0, LOW);
    digitalWrite(1, LOW);
  }
}


uint8_t generateNoise() {
  // Return 1 bit of noise using a Galois Linear Feedback Shift Register
  // See https://en.wikipedia.org/wiki/Linear_feedback_shift_register#Galois_LFSRs

  if (lfsr & 1) {
    lfsr =  (lfsr >> 1) ^ LFSR_MASK ;
    return (1);
  }
  else         {
    lfsr >>= 1;
    return (0);
  }
}

The RNG didn't help as much as changing the much-disparaged delay.

Are you sure the pins are correct?

Maybe use a smaller random number generator per https://arduino.stackexchange.com/a/18092/6628 ?

static uint8_t lfsr = 0x01;
const byte LFSR_MASK = 0x8e;

void setup() {
  pinMode(0, OUTPUT);
  pinMode(1, OUTPUT);
  pinMode(3, INPUT);
}
void loop() {
  //      int buttonState = digitalRead(3);
  if (digitalRead(3)) {
    if (generateNoise()) {
      digitalWrite(0, HIGH);
    }
    else {
      digitalWrite(1, HIGH);
    }
    //delay(2000)
    //for (int i = 31; i; i--)_delay_us(64516);
    _delay_ms(2000);
    digitalWrite(0, LOW);
    digitalWrite(1, LOW);
  }
}


uint8_t generateNoise() {
  // Return 1 bit of noise using a Galois Linear Feedback Shift Register
  // See https://en.wikipedia.org/wiki/Linear_feedback_shift_register#Galois_LFSRs

  if (lfsr & 1) {
    lfsr =  (lfsr >> 1) ^ LFSR_MASK ;
    return (1);
  }
  else         {
    lfsr >>= 1;
    return (0);
  }
}

The RNG didn't help as much as changing the much-disparaged delay.