You really need to get your actual tongue weight. Three passes needed as stated above. One with TV only, one with TV + TT and WDH hooked up, and one with TV + TT without WDH hooked up (bars in TV). With this, you can determine not only weight transfer to steer and drive axles, but tongue weight and weight on TT axles (before and after WDH hooked up).
Doing all the math with scale tickets in hand can be confusing, so ask for help if needed. I have been through this but am no expert.
You also should take your scale weight for the TV by itself and subtract it form the GVWR on your door jamb sticker. This will give you your actual available payload weight. Should do with full tank of gas. Can have driver only or family, dog and cargo as long as you take it all into account properly in the end. Important to know so you don't overload the TV. You want to know how many humans and pets and how much cargo you can put in TV with TT hooked up. You might get a bit of a surprise. Don't forget that your hitch receiver will have a max. tongue weight on it too.
Typically you run out of payload before long before you reach the GCWR or hit the front and rear axle ratings on the TV.
If you have a lot of bounce in the rear of your TT, you could have undersized spring bars. Your bars should have the max. tongue weight indicated on them (at least we do). You *could* have a lot more tongue weight than you think and if your bars are in fact undersized, it can result in more rear end bounce in the trailer. Just found this out the hard way ourselves. If your spring bars are too undersized, you will also have problems in "winding" them up enough to transfer weight to the steer axle.
I suspect you are not transferring enough weight to the front of TV. The number of links in your chains depends somewhat on what WDH you have. Perhaps you can say what it is and also post a pic. Links must be the same on both sides. If you find you have to use gorilla force to get the bars onto the snap-up brackets, something is not right. You should be jacking the rear of the TV up a bit with trailer hooked up to make it easier to hook up the chains. Do you know what your particular vehicle should settle back to when the WDH is hooked up? Different makes and models vary and your WDH manufacturer will likely say something on this too.
There is a difference in the number of links between trunnion style and round WDH spring bars, and again info. on your WDH would help. Typically you should have 3 or 4 links left over (hanging loose). Sounds like you have too many links between snap-up brackets and bars.
If you do the three passes you can determine how much weight is transferred to the steer axles, the drive axles and the trailer axles. Greatly assists in setting up the WDH more accurately. You could possibly have to adjust the number of links in the chains or even to have to tilt the bars up or down. You should get the trailer leveled better before you do all this though. Slightly nose down is okay so it sounds like another inch would work.
If you don't have any type of sway control, it is highly recommended, esp. for your trailer size. Various ways to accomplish this along with lower to higher cost.
Once you do get it all "dialed" in, you will find it is really nice to tow!