Well my heart rate while running is around 150 ~ 170, depending on the day.
Perhaps you are over working yourself or thinking too much while you are weight training or running. Try a relaxed approach and perhaps bring a friend along or some music to help you relax or instead of running at the gym, run at home in a place you are familiar with and try to go with a friend, family member, or pet. Remember, exercise doesn't require you to be in a constant state of anxiety in order to be of benefit. You just have to do "enough" (which you define and know) until you get the benefits of it.
Also, if you can't run at home, try this approach: for about a week, just hang out at the gym. Talk with people, don't do any exercise, maybe just look around, talk with some members, and just sort of be in a laid back state. This way, you get used to the environment you are in. Afterwards, try simple exercises. Maybe lift light weights, while still talking with other members, or perhaps you can walk at a very slow speed on the treadmill while talking with someone or listening to music. After a while, you'll stop thinking about the environment you're in, and think more on your exercise. Then you will feel better.
To be honest, when I'm running in a unknown place, I get a panic attack too. I tried one day at the beach, and I didn't run in the area before. I could barely reach a mile before I had to stop. However, at home, where I am familiar, I can run up to 3-6 miles nonstop without even thinking about it. Now I never say to myself that I'm going to run anywhere I don't know. I just run at home, or at my gym, and I can focus on it instead of worrying about the environment I'm in.
I think in your case, given what you said, it sounds like you are in an unknown place or perhaps you never really exercised before, and you just had a panic attack. Constantly checking your heartrate, and such, can further this panic because here's what is happening: you believe you are having a panic attack, you check yourself for the conditions you believe bring forth a panic attack, and when those conditions exist (whether real or not) you get into a panic state because your "conditions" are met. Therefore, you must remove the conditions that causes you to think this way. So, throw out your heart monitors, and just focus on being you at the gym.
