I would guess that you have "BIAS PLY" tires on the trailer rather than "RADIAL PLY" and with your cold weather the bias ply tires develop flat spots where they are parked. When you drive, it goes.... bump, bump, bump, bump with every revolution. The faster you go the faster the bump bump bump and once you get over 40 mph it feels smoother while under 40 mph feels worse.
This is just the way bias ply tires are in cold weather. It would take many miles at high speed to warm them up and make the flat spots go away.
And since you are new to towing, yes you will feel the trailer move you around and bounce you. It makes your truck bounce in a quick, jarring type motion. The kind you feel in your neck like whiplash. Not a smooth bounce like when not towing. Though sometimes you will get a forward and backward surging feeling if the load is heavy, like towing your tractor. You will get use to what is normal and then you can feel if something is wrong or loose.
Swaying is when the trailer starts to whip left and right, usually into the lanes next to you. This is VERY scary and very serious and will pull your truck with it. You never want this. This is caused by not enough tongue weight (weight on the front of the trailer) and too much on the back of the trailer. Swaying usually occurs over 45 mph (that is why uHaul trailers all have 45mph speed limit stickers on them). The next sway threshold is about 60 mph. Swaying is usually worse when going downhill and very hard to get back under control.
If you ever get into a swaying situation... Let off the throttle for sure but be very smooth (no sudden movements at all), then stay very, very light on the brake, too much braking will make it worse. But you do need to slow down and the preferred way to slow down is to NOT use the towing vehicle brake at all but instead manually activate the trailer brakes with the thumb slide on the controller. This will cause the trailer to slow down while you coast, this pulls everything straight and keeps the hitch tight. Also, be very light with the steering wheel, don't try to compensate as you will usually make the swaying worse. Just keep it going as straight as you can. If you ever get to really swaying far, you will probably take 1/4 mile to get it under control. You want to be very smooth and don't try to slow the towing vehicle down fast. A swaying trailer is trying to pass you and slowing down the tow vehicle fast helps it try to pass you. Usually resulting in jack-knife. Most important is to use the trailer brakes and then slow as fast as possible. With the trailer doing the braking you can put them on hard but don't lock them up. Activate the thumb control slow, don't slam it all the way. If the trailer skids, release and start braking again. Slow and smooth is the key to all of this.
Gene :^)
Note: I have towed my tractor forward and backward on the trailer. Doesn't make any difference so long as the load is balanced correctly for proper tongue weight.
Note 2: The ball size is normally stamped facing up on the coupler. Sometimes right on top of where the ball goes or just behind the latch with the other wording. Just look for a 2" or 2 5/16" They look quite different from each other even if it is not readable. Or just hold your hitch in your hand and put it up into the coupler, then lock it and see if you have a lot of play. The 2 inch coupler on a 2 inch ball will be pretty tight when latched. The 2 5/16 coupler on 2 inch ball will almost fall out or might fall out when latched. Will have alot of play for sure. I would guess that your dealer would not make a mistake in this area, it is very important and he wouldn't let you leave the lot with the wrong size.
Note 3: If the latch is the folding back type with the little lock lever, it is normal to take off or replace these. Look underneath and you will see a large nut to adjust the tightness. You may need to adjust this and should check it every couple years. Again hold your hitch in your hand and lock it into the coupler. If it is too tight, you take the hitch out and then loosen the nut half a turn. Then re-check the fit by locking the hitch back in. If it is too loose, you take the hitch out and then tighten the nut half a turn. Of course it doesn't have to be 1/2 turn increments, turn it whatever amount makes it fit good. You don't want play but you also want it to turn and move freely in there.
Note 4: Get yourself a can of grease with a plastic lid at the auto parts store and keep it with your trailer stuff. They are about 4 inch diameter and 3 inches tall. Then for regular maintenance before you attach your trailer you remove the plastic lid, turn the can upside down, and dip it (just barely) onto the top/tip of your towball. This will leave a small blob of grease on top of the ball. Then couple your trailer and the grease will get pushed around up inside the coupler and ball.
Always keep the ball greased and coupler adjusted proper. And always have a locking pin (clevis) or padlock on the coupler to keep it shut when towing.