Play your cards, use them, accept their destruction.
Equipment cards, as ongoing cards, are not meant to be permanent - they are here for some time, you take advantage of them, and then you lose them, often with a few HPs to round the deal. But it is worth it because while you have them, they give you the edge you need to slowly drive the villain to defeat.
If you have the luck to put together and maintain a good equipment combo, the hero will feel awesome - but none of these heroes is meant to be played with a big equiment combo everytime. It is not rare for Bunker to only play with one Equipment card available. AZ often has to play a long time without having his two module combo. They still can do a lot - but the contrast with what they can do with a big combo makes them feel less effective. It's like playing with Legacy : there is a big difference between a Legacy with 4 ongoings and a Ring in play, and Legacy without it - they almost feel like two different heroes.
AZ is the hardest to play without equipment. Self-damage is always hard to accept, and his base power is useless without modules. But keep in mind that you don't need the modules all the time. You'll need them to heal - so you can heal the self damage you deal yourself now later, when you finally get the modules to play with, so use your cards right now.
Most SotM fights, whatever the villains, are about rythm. The rythm at which you build your heroes to make them more effective, the cycle of buil-up / destruction / build again, the fast or slow inversion of advantage (from the Villain having lots of cards out at the beginning while heroes have nothing to the opposite), the rythm at which you can take damage and or resist it, etc. You take a beating in the first round. Then slowly fight your way back - until the villain gives you a beating again, and the cycle starts a-new.
I am in the habit of explaining to new players ( not that I suggest you are a "noob" ) that SotM, being a game about super-heroes, is built around the idea of sacrifice : to win, you have to accept to lose bits of your power. Be it your HPs, your cards, maybe even your character (incapacitated side is often useful). We , player , tend to think in terms of "leveling up" : you gain new, permanent advantage over time. SotM is more a game of small, temporary power-ups, slowly leading you to success. Unless you are very lucky and you can put together a big combo, like a fully equiped AZ or Bunker or RA, master of volcanoes, or Visionary, invulnerable ether-twisted demoralizer.
You have to keep fighting. Whatever the odds, whatever the frustration of having all your work undone. Even if you feel you can't play optimally - or powerless. Heroes never give up. 
I remember very well my first games of Sentinels - at first the villains seemed unbeatable. It took a lot of rematches to find how to survive and win, lots of discoveries. And lots of readjustment from a RPGer's mindset to a "whatever the cost, I won't let you win, Blade !" mindset 