First thing I would recommend is to step down your expectations. You mentioned no power or accuracy. That's absolutely normal when just getting started. Both of those things can take a long time to develop. You also mentioned getting 35 of 40 tennis balls out in front of you. That's great! Depending on the release type, it's common to have projectiles going up in the air, way off to the side, straight into the ground, etc.
As with power and accuracy, coordination takes time. Don't try to include too much body movement at the beginning. Work on getting comfortable with releasing the sling at the right time.
You should be able to dry fire any sling. But if you want one that will open up in a smooth motion (and possibly crack), you'll need a braided sling with a tapered release cord.
When I first introduce people to slinging, I usually start them off with an underhand release. Sling Moustachio has a good video showing this method:
https://youtu.be/I6kIZZQlW84. Something I'll mention though is that you don't need to rotate the sling as fast as he is. You can rotate and release it more slowly and controlled at first until you get used to the release point. And when practicing underhand, you don't need to permanently adopt this as the style you're going to use forever. This is about training your brain to recognize when to release the sling. Once you're doing it well with underhand, you can try out other styles.
I'd recommend sticking with tennis balls for at least a few sessions and see if you get used to them. No one loves them, but they're such a common projectile and no real threat to anything around you. Other options will be dependent on what kind of areas you have to sling in.