Hi ScantPalaver,
Your picture above shows one key thing I'm doing differently. I prepare the fibers before braiding.
See the appended picture. Do you see how the fibers I'm braiding with look like? And the raw fiber I put beside it? The fiber I'm braiding with is, well, fibrous. The fresh one is more of a sheet.
You want to remove all the soft parts and break the sheets somewhat down before braiding. To do so, pinch the fibers with your index finger and your thumb and pull them through a couple times. See the second picture. A lot of material should come away. Be generous and discard it. The fibers should break down into thinner strands and become soft and pliable. If you have too high quality fiber, you might have to use the fingernail of your thumb to help you a bit (and safe the skin of your thumb from burns...). Now you are ready to start braiding.
Regarding wet vs. dry: A wet sling is (quite often

) a dead sling.
The fibers will swell and away goes the nice tight braid you just made.
Regarding twisting: you can make very stiff braids without twisting at all. Not saying that it is a good idea though...
I guess your springiness comes mainly from the "sheets" you are using. Maybe also from a lack of tension while braiding. Hard to tell from a distance though... To help you, you can fix the braid to a post or something. In this way you can more easily maintain tension while braiding. You don't need to have tension on all threads at all times. Just on the one you are braiding with.