I think what you're doing wrong is expecting it to be sturdy like a building.
We have stab jacks, between the tire chocks, wood and plastic blocks all being used and still have just enough shaking to know which kid is getting out of bed on the other end of the trailer when we're laying on our bed in the morning.
I thought about putting in jacks by the tires too but people have said the wiggle is still present.
It's a trailer and as such you can expect some wiggle. My wife walks very heavy even though she is not that big, so if she walks you feel it. I on the other hand walk a little lighter so not much movement. Like others I do use X-Choks and they do help some but do not eliminate all the movement. I have tried using more blocks under the jacks but really don't feel much if any difference. I can see how adding jacks near the axles would probably help and if it were so bad as to bother me I would try it. If you think about it the tires will have some wiggle to them adding a set of jacks could very well help eliminate it to some degree. Bottom line is that it is a travel trailer so some movement is inevitable.
I got a pair of cheap slide room jacks and use them with a small block of wood on top right in front of the wheels. I put them in after I have the normal stabilizers down and the slide room out so everything is settled pretty much how it will stay since they do not move once set. There will always be some wiggle but they make for a pretty darn solid experience.
Last thing I need are the X wheel chauks for front to back motion.
We had the same problem. I recently added two BAL scissor jacks one just in front of the axles and one on the opposite side behind the axles. This all but eliminated the bounce in our trailer. It takes the tires and springs out of the equation. I would highly recommend trying this.
Couple things. If you are having to extend your stabilizers pretty far they will provide less stability. I carry 6x6x6 square blocks to put under each stab as needed so I don't have to drop them so far. Sometimes I'll even add more under those with levelers and such.
After setting up a couple things can happen that will cause you to have to adjust the stabilizers. Tires can change pressure and people walking and moving in the camper can cause shifts and settling that require further adjustments.
I too have set up and come back later to find a jack not touching the ground! Keep adjusting until you are satisfied. It will never be free of movement.
2010 Nissan Armada Titanium w/BT Mirrors
2010 KZ Spree 240BHS LX
Prodigy 2 Controller
Equal-i-zer Sway Control Hitch