Overwhelming Sound is Not Good for You

I'd be right there with you. Intense none stop sound can give you hearing damage and rattle your nerves. I strongly suspect that much of the overwhelming feelings is the worry that something else is wrong. Worry without relief has a way of spiraling up and away from reality, but it is very uncomfortable. Clearly, you have to do something and this is what I'd suggest.
Create yourself a hands on menu so your customers can point. Then get yourself a set of ear protection headphones. They can be quite effective and for under $25 you can get the best sound attenuation to 30db. Add a set of ear plugs if you need to and together that will cut sound considerably, so much so you will have to rely on your hands on menu and learning to read lips.  You can spend a little more and get music with your sound control. I bought a set to wear at the Indianapolis 500 this year and it was so quiet I was a bit disappointed at being under-stimulated and so less excited!
The struggle you may have with learning to read lips will be much less stressful than trying to tolerate the noise. Worst case, you may have to take them off for some customers until you figure out how to make it work.
You also may want to look into getting some breaks build into your day. If you have two locations, you must have some help. Trade off shifts if necessary.
Trust your body, it's telling you you must do something for relief. Find a way to make it happen. Good luck with your business.