Finally - a break-through!
The problem has always been breathing, and I finally figured out that my timing was off. Before, I would start my stroke with my right hand, then in mid-stroke put my head up on the left to gasp for air. But there just wasn't enough time to get that breath in.
I started timing my breath with my stroke, so as I start pulling through my right stroke, I'm already starting my left breath.
Bingo! Far more time, and as a result I get a better breath in and feel less panicked while doing it. I feel more comfortable, and at the end of my time in the pool I was able to concentrate more on form and less on not drowning.
I'm only doing 2 lengths (50m) between stops, but I think now that my breathing is working better I should be able to stretch that out longer. Maybe eventually I can get it to the point where it's continuous, then I can do a try-a-tri! 400m is a lot longer than 50m, and it's in a lake, but it's at least conceivable now.