Going over your numbers again, I have deduced:
Your trailer weighs 5960 lbs.
Your tongue weight is 780 lbs.
The weight of your trailer on the axles is 5180 lbs.
The weight transferred onto your trailer axles is 200 lbs.
The weight transferred onto your TV is 580 lbs.
Your children are young.
Unfortunately you do not have scale weights for the steer and the drive axles separately. The weight onto your trailer seems pretty good and you *may* have appropriate wt. transferred to your front axles. The only way to tell for certain is getting separate axle weights. If the front fenders of your TV have settled down to near the height without trailer hooked up, I am thinking you are pretty close.
You want to end up with your round spring bars close to horizontal so you may want to adjust the number of washers in the shank head as needed. Hard to say how many washers equate to the number of chain links, but I would think it you measure the height of your fenders, you will end up being close. Only other thing that I thought of is that with each pass through the scales, you want to have the same number of people, pets and other stuff in the TV so you are comparing apples to apples.
No one else has jumped in so I hope I am not too seriously wrong!! :)
Happy camping.