Adjusting simple (so it should be) shift register coding for a different result

  for (int thisLed = 0; thisLed < ledCount; thisLed++) {
    pinMode(ledPins[thisLed], OUTPUT);
   servo.attach(servoPin);
  }

How many times do you need to attach the servo?
How many times do you need to attach the servo?
How many times do you need to attach the servo?
How many times do you need to attach the servo?
How many times do you need to attach the servo?
How many times do you need to attach the servo?
How many times do you need to attach the servo?
How many times do you need to attach the servo?
How many times do you need to attach the servo?

when 'angle' is between around 94 to 113, led pins 13,12,11,10 are all lit and should be off.

A value of angle of 95 or more means that the potentiometer reading was 570 or more. That means that ledLevel will be 4 or more.

  for (int thisLed = 4; thisLed > -5; thisLed--) {
    // turn the pin for this element on:
    if (thisLed > ledLevel-1) {
      digitalWrite(ledPins[thisLed], HIGH);
    }
    // turn off all pins higher than the ledLevel:
    else {
      digitalWrite(ledPins[thisLed], LOW);
      }
    }

When thisLed (dumb name) is an index into an array, it makes NO sense for the range of values to be anything other than 0 to ledCount. You are iterating with thisLed equal to 4, 3, 2, 1, 0, -1, -2, -3, and -4, and then reading from 4 undefined positions in the array.

Since ledLevel is 4 or more, ledLevel -1 will be 3 or more. So, you are turning the pin at position 4 in the array (13) on. You are then turning the pins at positions 3, 2, 1, 0, -1, -2, -3, and -4 off. Since the pins at positions -1, -2, -3, and -4 are undefined, who knows what pins you just diddled with.