We use foaming hand soap here, and even that can spit out at you. I don't know anything about foaming dish soap, but I imagine the shooting straight out thing may still happen.

Dawn makes a foaming dish soap dispenser. But it sounds like you want it installed in the sink, and this would have to sit on the counter.

Dawn foaming dish soap

I'm not sure about the foaming dish soap dispenser, but what I do know is that it's more about the design of the dispenser than the ingredients. You can make any hand soap into a foaming hand soap based on the dispenser, such as in this recipe: https://www.getgreenbewell.com/foaming-hand-soap-recipe-using-essential-oils/ Perhaps there is no foaming dish soap dispensers out there. But the detergent you put in will likely be the same as in a normal soap dispenser.

Also, I've had MANY problems with soap spitting at me. For me, it was always in hand soap dispensers. And it was when the hand soap gunked up in the dispenser. I found that castille soap was among the major offenders for doing that. If by chance you're DIYing your soap or if you are using a natural or organic soap, there might not be an ingredient in there to keep from clumping up. So when it comes down to it, a foaming dispenser might not be the 100% solution that you're looking for (although that would be great!).