Actually you will get LESS sway with MORE tounge weight. If you reduce tounge weight you will get more sway.
Controlling Sway Causes of poor tow
Tips on How To Tow a Trailer | etrailer.com
I tried the scale method but most bathroom scales will only go to 300lbs, so at 4.1 you will max the scale at 1200lbs. I used a longer bar so went with 6.1 but still ended up bending the bathroom scale.. Eventailly I went into a closed roadside comercial scale and weighed the truck empty, then the trailer tongue (un-hooked) I weighed truck front rear and trailer connected. My toungw weight was 1350lbs.. Gross Trailer weight was around 12,000. I was shooting for 10 to 15% tongue weight and was pretty much right where I wanted to be. That said havng a older truck my OEM hitch was rated at 1000lbs tw and 10,000 gross. I upgraded the hitch to a
1600 tw 16000 gross rating.
I would suggest that OP also check the rating on his hitch to make sure he is not under rated but eventially I would get a bigger truck...