How do I know I've got the right amount of moisture in the paper to wet-fold origami?
The amount of water to use will very much depend on the kind of paper you use. Thick watercolor paper can absorb a lot of water and still stay intact while copy paper will disintegrate or at least tear with even a little moisture. This behavior depends on the paper's thickness, fibre length and sizing. Most wet folding tips found on the web (e.g. about using a spray bottle) apply to thicker paper and not to copy paper. Personally, I have never been able to wet fold copy paper and be happy about the result - it comes apart with even very little water.

So, first of all, I suggest you get some watercolor paper, 160-300 gsm for a start. Such papers are widely available in artist supply stores and not terribly expensive. Canson Mi-Teintes is a popular brand that comes in many colors but others should do as well. Then, retry your folding sequence and you should see a huge difference compared to copy paper.

The exact amount of water to use is unfortunately a matter of practice, but when I started wet folding, I always used too much rather than too little. It is easier to re-moisten paper than to dry it quickly, so better start with a smaller amount. You have to remoisten as you fold anyway. For most models, your paper should be damp but not soaking. While 160 or 300 gsm paper feels very stiff and almost not foldable at all when dry, with the right amount of water added it should feel leathery and be able to roll along a smooth curve easily. For some models, you only moisten a certain area of the paper at a time rather than the whole sheet. Also note that moisture takes some time to penetrate the paper, so you should wait a few seconds before applying moisture and checking if the paper has the right properties.

The rest is a matter of experience, so start with thicker paper and give it a try once more.