Is the noseweight to high making the rear of the car pitch a bit or the cars's rear shocks worn.
Also check the hitch damper and as 'littlejack' says have the brakes fully released. Was it the first trip away after winter storage? There might be some corrosion on the drums which will take a few miles to wear off.
If you can feel a bit of a thump as you move off & after braking, check hitch damper. With caravan uncoupled try to push & pull hitch in & out, compressing rubber bellows. It should be very hard to move. If it slides easily in & out with no resistance the hitch damper will need replacing. You can diy if you are ok using tools.