Looking at the trailer from the side, it appears it does need more tongue weight. I have no idea what the trailer weighs, but the rear of the trailer is high, raising the center of gravity. First try and put more air in the tires. On highway running near the 40-50 pound area will not hurt you in any way. Off road you normally are not traveling at speeds that would show the sway.
I use a very simple formula for placing the axle. I am not concerned with the length of the tongue in this. I take the box. If the box is 10 foot long, I center the axle 10 inches behind the center line of the trailer. 1 inch for every foot of box. It has always worked for me.
You have fantastic skills, you should be able to move a spare tire under the front or mount it there if needed. Still think higher air preasure may cure the problem. Also look at lowering the tongue connection at the tow vehicle. Normally the front of the trailer should be 2 inches lower than the rear of the trailer. Draw a straight line out from the back and measure from there, due to the anlge you have on the rear.
A trailer is basically a teater-totter. You want the most weight on the front.