Here's a link to a calculator that might help.
There are a number of things that can cause trailer sway, main causes listed first:
Overloading, insufficient weight distribution, insufficient tongue weight percentage, towing trailer nose up, unbalanced tire pressures, tow vehicle tire sidewall flexing, mechanical (bearings, axles, struts, springs, etc) failures. If you have a sway issue, it's possible your cause is anywhere in between just one item (listed above) to a combination of all of them. Getting accurate weights and ratings on truck (curb weight, GVWR, GCVWR, and tow capacity), trailer (GVWR, actual gross weight, actual tongue weights) and hitch equipment (weight rating) is a good start. You already know, tongue weight is not a constant number, so you need to plan on it being it's heaviest.
I've heard toy haulers run heavy on the tongue (average travel trailer runs about 12 - 13 percent) to compensate for toys in the garage. So, I used 15 percent tongue weight to calculate your approximate gross weight and based on 1500 lb tongue weight I came up with 10,000 lb loaded trailer weight. That's a lot of trailer for a Tundra with 4.7 V8. Even if your Tundra's advertised max tow weight rating were 10,000 lbs, once you add a passenger or cargo (like a bike in the truck bed), it no longer has 10,000 lb tow capacity.
There are band-aids that might help with comfort, but, there is nothing you can do to increase your tow rating.
As far as moving things around in the trailer, if you get your tongue weight percentage too low, it will increase your sway issues.
Check your truck tires for a good PSI (E rated LT tires should be good for about 80 PSI). If your running lower tire pressure, you could have a little bit of sidewall flexing. Make sure they are equal in pressure. Check trailer tires for equal PSI readings
If the truck doesn't have a sway bar, maybe adding one would help with truck body twisting.
It's possible your WDH is not restoring enough weight (it's primary function) to your front axles. If your front axle has lost too much weight, you loose some steering control and create more of pivot point at the hitch. One of the main causes of trailer sway. BlueOx Sway Pro has WDH models rated for 1500 and 2000 lbs.
I too, think you may be pushing the limits on your Tundra.
