I've been to the ER three times (as an adult) for panic attacks and each time my vitals did very similar to what you just described. That's why I stopped going to the ER! When you get there you are so ramped up and stressed about the reality of what's going on that your pulse and BP are through the roof, but your body can't keep that up forever so it slowly comes down enough that they let you go home. One time I had to be carried in by two adults because I was shaking so badly I couldn't stand up on my own.
You can help speed this calming effect by using biofeedback and relaxation techniques, but different techniques work better for some people than for others. The most immediately helpful thing for me to do is to take a deep breath, as deep as I can, hold it for a moment, and then blow it out slowly while imagining that I'm blowing out all the fear and anxiety. Do this a few times and you will start to feel immediate benefit. It's not as long lasting as the benefit you can get from other techniques, but it's fast and easy to do under any circumstance so I recommend trying it.