Unload or keep on truck

If you use your camper off the truck and want to brace the front jacks to prevent movement look at equipment like Torklift Wobblestoppers or Stablecampers or other such brace systems.

Loading / unloading a TC alone is not fun for the unexperienced.
I would leave the camper on the truck unless you absolutely need to unload (for example if the truck needs to go to the shop).
Always pick a level area of sufficient size for loading. You need enough maneuvering area to back the truck under the camper as straight as possible. The ground should be firm and level. Maybe you can find someone to do some practice runs with.
If you do unload... Many people lower the jacks as low as they will go. That reduces sway. Others use some form of stand to set the camper floor on. You do not really need to do anything, but with the jacks extended and no additional support, there will be some sway - which also puts unnecessary stress on the jack mounts.
If you are not used to driving with a camper, you need to stay aware of the dimensions of your rig - especially height. It may make sense to put some note inside where you can see it that reminds you of the actual size of your rig.
If you have no experience loading / unloading, it may also help if you write down a checklist of all the things needed to do it right - are the turnbuckles in place and tight, do the lights work, are all windows and roof openings closed, is everything inside safely stowed, are the stairs up/stowed, things like that. It is also helpful to do a walkaround some distance around your rig before you leave in order to check if everything is ready for departure.
Nothing of this is difficult, and it gets easier with every time you do it. But if all of it is new to you, it can stress you out big time. It is not rocket science, you can and will do it!

Sounds like no need to take it off the truck since you are towing a car. You should practice taking it off and on but that can wait until you have a nice open level spot to practice and are under no need to rush. People have different sensitivity to rocking. When we have our camper off the truck we don't find any objectionable movement. Others may have different sensitivities. I suggest that when you are able to have it off the truck, try it out and see if you are bothered by any movement you might experience.

since you’re towing a car you won’t need to off load your camper. But if you plan on keeping it you should practice on and off loading it. Being comfortable with the process will eliminate the need to tow your car so you can use the truck. Another option is to find someone local who can help you. Maybe the previous owner? Good luck