You are probably going to get a few schools of thought on this, but mine and most of the rest of the carriage world is that the wrap straps (if that is the form of "tie downs" you have on your harness) should not be tight enough to transfer the motion of the cart to the horse. I have also seen too tight wrap straps and overgirths (other form of "tie down") gall the horse in the elbow. It happened to our horse (again) a couple of weeks ago when a clinician adjusted our harness and felt that the overgirth should be tighter. When we were done, whalla, elbow sore.
(I would like to try a padded girth next. $$)
Now that being said, carriage drivers tend to use breeching to control the slowing and stopping of the cart. Without breeching or thimbles, wrap straps are often used as "brakes", and therefore tend to be put on tighter than I would do.
The actual purpose of wrap straps is to keep the cart from flipping up over backward. Without them or the overgirth strap, the cart shafts could raise up in front in certain situations, dumping the driver out the back.
Your shafts should not be bouncing up and down regardless of how the wrap straps are adjusted. If they are, that means that the balance of your vehicle is not right. Some people like 0 weight in the tugs (shafts "floating"), however, I tend to like a little, just enough so the cart doesn't bounce. Think about it, riding horses carry weight all the time, weight on their back is not an issue. Too much weight on their back is an issue. The cart bouncing up and down on their girth is an issue (annoying). Try raising and/or lowering your tugs (what some people call "shaft loops") and see if that helps the bounce. Also, adjust your position on the seat. You may be sitting too far back.
Another good exercise is to have someone sit in your cart while you hold the shafts where the tugs go. Have the person lean back and forth and see what that does to the front end of the cart.
It can be a big "ah-ha" experience.
Myrna