I go kickboxing for an hour a week, it is more of an intense workout than martial arts, although we are learning the moves and stuff. I didn't feel so good after last time, but only because i don't think i am as good as my master would like- but then i have only just started. If i didn't exercise in a structured environment without somebody telling me to do this and that i would probably fall down on the floor and decide to sleep instead. I can't do all the things that they say but i think the fact that i try is enough.
You can maybe make some improvement by not stopping exercising abruptly, wind down slowly, so that your breathing starts to return to normal before you have stopped, and try to work on breathing more during the exercise. I think if your breathing is better before you actually stop, and if you have a towel and bottle of water or have a shower straight after it might help a little.
I was in class once, and it was a small room, and very hot and crowded, the lecturer brought attention to this fact, stating that some students become quite faint in those conditions so to put your hand up if you feel that way, and half way through the lecture i was feeling like i was having a freak-out, it became very triggery very fast. I think that may be similar to what you felt.