Once you've addressed the WDH itself, there are some other factors that affect sway. The trailer should be level to slightly nose down. Tires on the TT should be at the max. sidewall psi. Running higher pressure on the TV tires can help when towing. If you don't have LT tires, they will help and running higher pressure on those too will help. Heavy duty shocks on the TV will help keep the TT + TV more stable. On a 10 year old TT, you might want to inspect the suspension. I think at the least you will find worn out spring bushings. Not sure what an '08 Spree has but you might benefit from installing some good equalizers like Dexter EZ-flex (and their wet bolts) or the Mor/Ryde CRE3000.
It looks like the Spree 260 you have has a dry weight of 5500 lbs but the GVWR is probably 6800 lbs. Always ignore dry weights. We have a newer version of the same Spree TT. It weighs close to the GVWR with camping "stuff" for the two of us and with empty holding tanks and the tongue wt. is close to 1000 lbs. I highly recommend going to a scale to get the actual gross TT and tongue wts. If you switch to another WDH, you want to ensure you have the right bar rating for it. You may find that the Ram doesn't have enough payload capacity for the actual TW + camping stuff in the bed + passengers.
I think a step up in WDH type/design like Equal-i-zer, Reese dual cam and Husky Center Line will help a lot for the Spree 260. We have the Reese DC and I love it and have zero sway. Semi trucks come and go on the freeway all day long and have no effect.
FWIW, shocks on the TT will help. We installed them and they really helped the overall control of sway and also improved handling on bumpy & twisty roads.