Desensitizing was the right idea, but either you didn't see it through or you didn't do it correctly.
Correct your own behavior
I'll take a wild guess and predict how you react to a whistle:
- You walk with your dog
- You hear a whistle somewhere
- You instantly look at your dog to see how he reacts. Maybe you even stop in your walk.
- You talk to your dog in a calming manner to tell him he shouldn't be afraid. Maybe you offer him a treat to calm him down.
Now let's run through the same scenario from your dog's perspective:
- I walk with Aron
- I hear a whistle. I'm highly alerted because I learned that this is a warning sign. (Maybe something bad happened in the past)
- Aron's body language gets rigid, so he must be alerted by the sound. He stops as well and looks at me. He confirms that I have to be alert. He wants me to protect him!
- Aron talks to me but I don't understand what he says. He must try to convey that this is an extremely dangerous situation. For a reason I don't understand, he offers a treat, but I must stay vigilant to fight or run away.
You see, trying to calm a frightened animal down often has the opposite effect. You reinforce the unwanted behavior.
Instead, you should model the behavior you expect from your dog. If you want him to ignore any whistling, ignore it yourself. Don't react to it in any way and don't look at your dog. Keep your body language loose and relaxed.
Desensitize your dog
This won't work in a day or even a week. Playing the sound of a whistle on a computer is more likely to reinforce his fear because you are more concentrated on your dog than usual. You are agitated when the whistle sounds, so the dog gets more agitated himself.
Instead, you should create a relaxed and fun environment. Play his favorite game with your dog. When he has the most fun, blow a whistle out of the blue and without warning. Continue playing as if nothing ever happened.
It's beneficial to not let the dog see the action of blowing the whistle. If you play with another human, let the human your dog is facing away from blow the whistle. Instead of blowing a whistle, you can play the sound from a smartphone or similar, but don't warn your dog by letting him see you taking out your phone and looking at him before the whistle sounds. An added advantage of playing the sound on a phone is that you can start with a low volume and gradually increase the volume over time.
Start with no more than once a day and low volume, but do it every day or at least every second day. Then gradually increase the intensity of the trigger (in your case: whistle several times a day and gradually louder). You can expect your dog to start ignoring the trigger after a week or two and maybe ignore whistles completely after 4 - 6 weeks.
Here is a great step-by-step instruction by wikihow.pet how to treat fearful dogs in general.
This video form BrightDog Academy perfectly sums up fearful behavior and the treatment of it (the most important part for your particular problem is the start until 8 minutes in)
In the meantime
In your comment you mention a very important problem:
Normally (when not in panic) if he starts pulling really hard on the leash, my reaction is to correct him (sit, stay, etc). Problem is, when he is in panic-mode this means I have to correct him nonstop until we are far enough away. But ignoring such pulling also seems like an "unnatural" reaction from me. So in this case which is better, correcting him for "bad" behavior or simply acting like he's not pulling insanely?
No you should not "correct" his behavior when he is panicked, because there is nothing to "correct".
We "correct" unwanted behavior to train our dogs to act in a way we like. We teach them "if I say 'sit down' and you press your bum on the ground, you get a treat" or "if you pull on the leash our walk gets disrupted by obedience training".
When a dog is fearful or even panicked, these rules don't apply anymore. The dog cannot think calmly and therefore cannot learn anything from this experience. He is not misbehaving but rather expressing his fear and this expression simply doesn't coincide with "good behavior".
If you encounter a trigger on a walk and it's a single whistle that will not repeat, stop and wait calmly for your dog to come out of his panicked state. He can still be agitated, but he should not pull you with the leash.
If you are walking and the trigger is likely to repeat (because there's a playing field nearby), change direction to walk away from the field and then stop to calm down. If (at the beginning of the training) your dog can only calm down back home, then you go home. You should desensitize him in a calm environment where he feels safe before you can expect him to stay calm on a walk.