IMO with a trailer that long you can put all the X-chocks/Rotor-chocks, JT strong arms, etc. on you want, but they will basically be wasted $$$. As mentioned your movement is coming from two main sources ... the trailer suspension/tires and from frame flex over the long spans between the ground supports (i.e. the sloppy stabilizer jacks at the corners and the suspension somewhat in the middle). The only real way to get the proper stability is to get some addition "GOOD" support closer to the axles which will help eliminate that source of movement and reduce the frame "span" length and help reduce the frame flex. I'm very much in the "Stabilizer Jacks" hate group and actually have removed the OEM ones from the front and back of my trailer. I now only use 4 of the stacking screw old school "jack stands" on our 31' trailer and when properly tensioned along with the tongue jack I get rock solid stability except if I'm walking around "heavy footed" in the very rear of the trailer. However our bedroom is in the rear so excessive movement back there is minimal so I didn't so far feel a need for proper support at the end of the trailer to eliminate that source of "residual" movement.
My recommendation is to get some "proper" support about 4' in front and in back of each axle and get those tensioned properly. You need to get that support done first and then your factory stabilzer jacks at the front and back might be able to be tensioned enough to provide the additional support of get your trailer "stabilized" properly. However, I'm somewhat skeptical about getting enough tension on those existing stabilizer jacks w/o using the tongue jack and that won't be an option with the screw jackstands already installed and tensioned. It's probably worth a shot, but don't expect very good results.
Personally for a trailer that long I would just remove the existing stabilizer "JUNK" and use 4 of the screw type jackstands near the axles and then use 4 lighter weight bottle type jacks at each corner so those could be "properly" tensioned after the 4 screw jack stands have been tensioned using the tongue jack. In fact I would probably experiment with placing the 4 bottle jacks like 2 or 3 feet in from the frame ends to even further reduce the unsupported frame spans to the screw jack stands. You don't need HUGE bottle jacks and the small ones with around a 2T capacity should be plenty large enough. The reason for using bottle jacks is that you can tension each one against the frame individually w/o having to use the tongue jack to "leverage" the frame.
Yes you're going to have to install and remove these items each time you stabilize, and it might take a few minutes longer, but you also won't have all that additional hardware "dangling" under the trailer subjected to the road grime, etc. It just all depends on what your desires are and any trade offs you are willing to make. I would guess you could get all the extra hardware I have mentioned for probably less than the cost of either the JT strong arms or a couple of the fancy X/rotor Chocks.
Larry