Hey, Jax main here, and I can understand your pain, especially if you are an ADC main (Jax is pretty specialized into hunting ADCs). Jax is designed to be THE melee hypercarry, so his damage is really oppressive once he hits his late game. Here are a few tips for winning fights against Jax, because there may be more counterplay than you might originally think:
Jax's kit allows him to deal hybrid damage, and thanks to his ultimate, gain resistances for building AP (MR) or AD (Armor). However, AP Jax builds are uncommon compared to his AD builds, so if you want to kill a Jax, itemize your build to enhance any magic damage abilities your champion has.
However, if you still want to play an ADC and mess up a Jax, I recommend Corki. He has an escape from Jax's Q, deals lots of magic damage, and has the ranged abilities to harass a Jax out of lane. Regardless of the champion you play, though, make sure to stay near your support. They will give you the peel to survive a Jax, and if they're a tank, they can also provide the CC to ruin his day.
Jax's damage is almost exclusively single target. While this mean his dueling power is unparalleled in League, his team fighting ability is rather unimpressive. Working with your teammates to bring down a Jax early in a fight is crucial; he is a melee carry capable of taking down your whole team (especially once lategame hits), but only one at a time, so treat him with the same priority you would an enemy ADC. AOE damage is a Jax main's nightmare.
Wait until the fight is halfway through to use your CC. If Jax can't autoattack you, then his stacks fall off really quickly. Using your CC halfway through a fight means he has to start building up passive stacks all over again, thus reducing his damage substantially. Your team's tank is your best friend when you are playing against Jax.
Like you said, your auto attacks do nothing against a Jax with E, so run when you see that lamppost spin. If you can get adequate distance between you and Jax, this is the time to use your abilities. All of Jax's abilities, aside from his Q, run on timers. His dodge ends. His empower eventually drops. His ultimate's active has an 8 second duration with a minimum 70 second cooldown. Furthermore, his ultimate and E are not cheap mana-wise, nor do they have low cooldowns. Try to bait the ability, and then engage once the particle effects fade or the lamp stops spinning.
I have more tips, and I'm sure you have disagreements, so I'd be happy to talk through any concerns you have.