Yes, Oats are cereal grain. If you are sensitive to oats, then eliminate grains, including corn. Again, corn is not a vegetable. Corn is a grain.
Oats can increase your glucose high. It can also make you feel hunger a few hours later. Really. I did self-experiment myself, and did look up info. Many people feel the same way. I don't have gluten problem, but it gives me sinus problems, and inflammatory. So, I had to eliminate them, and I saw the major difference. No inflammatory. My nose doesn't clog anymore. I can breathe through my nose. It also reduces my allergies,too. I did not even take allergies medication. And the rest.
You can try different types of meals without any grains. It is not the end of the world.
I recommend avoiding dairy and grains for three weeks to see how you feel. After that, you can re-introduce them to your body again. Wait and see if your body reacts. Write food journal. You have to get to know your body better.