Full time means your motor or transmission or something related to the driving part will have to go into the shop at some point. Even mechanics cannot do many things "on the road" due to not having the tools or shop as if they were at their home garage...
A trailer can be parked and then the tow vehicle, usually a truck, but suv also, can be driven like a regular person drives, or towed to the shop.
A truck can be towed by any tow truck, and a trailer can be towed by any tow truck, and sometime the driver can use a flatbed for truck and tow the trailer behind the same tow truck (ask me how I know).
A class C, even a small one, CAN be towed, but most large tow companies refuse it or have to bring in a very big truck, usually a big flatbed, and drag the motorhome up onto a 4 foot high platform and drive it to the shop.
So, harder to get a tow truck for a motorhome than truck and trailer. I had to sit in a parking lot for two days trying to get a tow truck to pick me up once (I had a truck camper, which is a motorhome as far as towing is concerned).
As others mentioned, being able to drop the trailer and go drive around, especially in some national parks, is a huge plus. Full timers will end up parking/staying in one spot for longer periods than vacationers, so likely you plan to have safe spots to be able to just leave the trailer and go shopping or driving without your home behind you.
Vehicles, even motorhomes, lose value with age and lose value with every additional mile you rack up. Trailers hold value based on their condition, and age is a smaller factor. Another vote from me for a separate trailer.
Towing a car behind a motor home means now you have two vehicles to keep up, two vehicles to insure, two vehicles to pay taxes on, and a car is heavy, even a small one so will ultimately put the motorhome into the shop sooner than later from the stress of towing it.... If you are towing the car on its own wheels then that car will rack up the miles even though it is not running, and those miles count the same to a buyer if it was driven or towed. Possible to put 100.000 miles on a car that has never been driven this way. Maybe some cars do not count miles if the vehicle is off, I do not know for sure... If towing the drive wheels on a tow dolly then you avoid this, but you cannot back up a tow dolly with a car on it more than a few feet. Total pain in the neck, and you do not want this full time. Another vote for truck and trailer, one vehicle.
With a motorhome you can panic and drive away, but a trailer you have to get out to get in truck. If you want to live in RV, you can't be this sissy. Reality check, escape is not something to consider, it is just not something very high on the list of reality. If someone is after you then that glass drivers side window is not going to protect you either in he time it takes to start up and put it in gear. You can't always escape an attacker.
Better to come up with self defense protection you can have from your bed.
When the vehicle wears out and the trailer is still in great shape.... trade in truck for new truck and drive away with your old trailer....
Also, get the smallest trailer you can live with. Big giant 5th wheels is great for going to an rv park for a vacation, but the reality of driving with a trailer all the time is that a smaller one is easier to drive, fits in parking places easier, turns around easier, is easier to maintain on the outside, is lighter so less wear on truck, fits in places better (you will not always be in a campground, maybe a friends driveway and they will appreciate a small rig), is lighter so less wear on the trailer tires (less trailer tire expense), can be towed by said tow truck driver, and will give you better gas mileage so you feel more free to roam.
Narrow is better than wide. 7 foot wide is better than 8 foot wide. mpg will benefit. Narrow is easier to see behind with mirrors too.
Same goes for trailer height.
I suggest a tow vehicle with a lot of storage. You can store stuff here, and store stuff that you can take with you if you do drop the trailer off and go drive without it for any reason.
I have had, and traveled in a lot all of: one Class C, two truck campers, and 3 trailers. I vote for trailer.