Why don't I get the symptoms most commonly described, like clammy hands and a tight chest? Is it my adrenal system or something?

Everyone's bodies are different and everyone has different reactions to stressful situations. That's why people can't tell how others are lying based off of if they blink three times, or look to their right. Everyone's different and everyone's bodies copes with different stressful situations in different ways.

Sometimes I don’t get symptoms either and I didn’t at first, I’m not sure why this happens but they are just symptoms and symptoms aren’t always a definite thing that happens

Anxiety manifests itself in many forms. Clammy hands and tight chest are just 2 common symptoms but are not the only symptoms. There are many symptoms like shutting down, feeling numb, fidgeting, lost in train of thought, afraid of doing things that may cause embarrassment, sweating, avoiding answering in fear of being wrong or sounding inadequate, overthinking, overanalyzing, making up many scenarios of a possible outcome before responding......and the list goes on and on. Anxiety is subjective to the person. You will know yourself. What feels right, what feels wrong, what feels off etc. You are the only one walking in your shoes. Basically you are 100% orginal and so don't use anyone else's symptoms as a benchmark for how you feel. Learn to be in tune with your own body.