I think mostly everyone has covered what I would tell you. Yes, this is sort of typical, not just of basenjis, but of many dogs that tend to "stress down". Zooming would be stressing up, been there too! The Control Unleashed book is what really helped me understand a lot about stress in dogs. And yes, I've had bouts of Zest! having "stage fright" for lack of a better word. Sometimes Rescue Remedy will help, but it's really only part of the solution. You do need to expose Gossy to many different place to do agility (or even just basic obedience). Hopefully you can find someone local who has some equipement in their yard that you can borrow. Or perhaps just taking a couple of jumps to somewhere else may help. I think it's important to end the session before your dog quits on you especially in a trial situation. About a year ago Zest started freezing in the ring. So, I travelled 2 hours in the winter to a trial and our first run I did about 1/4 of the course. And celebrated like we'd just gotten a MACH on the way out. Z likes it when I'm animated. It took a few trials before we were doing the entire course, but we got there. (And she was in Excellent by this point, so it wasn't like it was her first few trials.) This weekend on her first run, she ran out of the ring after about 1/2 the course and when I went towards her, she ran up into the stands. She's never done that before. Her 4th (and last run) of the weekend (sunday) she did a beautiful run with only one R (so, we didn't qualify, but it was a HUGE improvement over the previous day).
Also, keep a book. Note place, date, course and anything else you think might have an outside chance of affecting Gossy. I have place, date, time of arrival, time of run, temperature, course, if I used RR, if we spent the night in a hotel, and write down what she did/didn't do and anything odd that happened. You may start to see trends or, hopefully, improvements over time.
The other thing you may want to do if possible is to enter a jump height lower when you first start competing. I entered Z in Novice Perferred the first few times. Then I'd entered her in regular Novice, just to get as much ring time at the novice level as i could. Even so, I may (or may not) have rushed things a little. Yes, i know Gossy loves jumps and jumps easily, but that's not really the point.