As others have said, the bumping you feel is the transmission downshifting. It should not be shifting so harshly that you feel it really applying the brakes, unless it has been tuned to do so. Check to make sure you're not driving in a sport mode. The drive to reverse harsh engagement is a sign of wear. It definitely should not be so rough. Are you coming to a complete stop? Seems silly to ask, but people are hard on cars, especially if they are new to the category/performance vehicles in general.
Gen 2s are kind of floaty, and you say it's your first truck. They're also body on frame trucks with a stout suspension, so some bed shake/shimmy over sharp bumps is to be expected. To make sure you're not just experiencing the physics of a truck for the first time or just the quirks of a Raptor, see if you can drive another Raptor.
Airing down the tires will make the truck less jittery unloaded, but it will also make it wallow more around corners. Personally, I like this, as it makes it more baja-truck like. Note that it will be less responsive on the street at lower tire pressures. You may also activate a low tire pressure warning.