Welcome to Dallas!
I encourage you to talk to several different resources. ADHD is a big and profitable business. Schools , especially private schools, want kids "in the box". If they are not , then they want them medicated or OUT.
We were told in PreK ( age 4) that our son should be tested. We spent three long years trying to figure out "what was wrong".I had him evaluated at Our Childrens House at Baylor (an excellent facility, there will be a wait, but get on the cancellations list and you'll get in a lot faster.)
I got a nutritionist involved which was probably , in retrospect , the best thing I did. Changing his intake of gluten and refined sugars was good for all of us!
In the end we had him do occupational therapy in second grade as he did have some sensory integration issues ( check out Lakewood Pediatric Therapy for this) which mimic ADHD. Now that he is 11, in hindsight, I probably would have opted to quit working, be more hands on and get him into more outdoor activities. ( In other words, do the occupational therapy myself!)
Most kids today don't get the physical activity they need and that is one of the reasons we see so many manifestations of sensory and attention issues.
But lets be realistic. It's PRE K. Kids are not SUPPOSED to be paying attention and listening to teachers. They are SUPPOSED to be playing in the grass, counting clover and wrestling with their pals. They are KIDS. I dont' know about you but at 4, even at 5 I was not in a classroom, I was playing, outside with my mom and my friends. It sure did not hurt me down the road when I graduated with honors!
I'm an older mom and my expectations of the service an academic atmosphere should provide are highly colored by my own excellent experiences growing up in Department of Defense schools. My husband had equally excellent academic challenges in boarding schools. We finally gave up on schools, both private and public, completely, and now homeschool to the delight of our child. He is thriving , works a full grade level above his peers and has a more active social life than ever.
While not everyone can homeschool, you can take a look at different learning environments and maybe even hold him back. Pre K is not a necessary experience for children. They do not NEED to be socialized. While it can be a great experience for some, it can be a disaster for others. One triplet may love it, one may hate it. Each child is so different. I encourage you to listen to your heart, only you know what is best for your child.
Scottish Rite Hospital is another excellent resource. I can certainly second the advice about Shelton School and the others you've been told about. Dallas is full of resources and one of the best places in the country to live if you truly find you need to seek treatment for a spectrum disorder.
Good luck and whatever you do and don't let it get you down.