Having been camping all my life, the father of 2 children myself, traveling with mother-in-law and mother, and owning my own campers for a long time now, and having done a lot of traveling, I will say from experience, that the advise and information that everyone who posted here are 100% correct. Especially the advise about the time limitation you have remaining to locate an RV and to work the bugs out of it before venturing on your trip, and the availability of campsites.
However, it's still your trip. We can only give advise from our own experiences. In the end, it's your call what decisions you make and how you do it.
One thing to remember ... there really is not "right" or "wrong" way to go camping. We all just do it slightly different. The MOST important thing is, to drive the highways safe! If you settle on camper, you may find your F150 turns out to be a "white knuckle" drive if the set-up is incorrect. It does take time to iron out the bugs. You will not be able to travel as fast towing a trailer. If you don't spend time and really get to know the camper and tow vehicle limitations before you head out, it could be a disappointing drive. Far too many on these forums start out with F150 or 1500's and then have terrible towing experience (for a multitude of reasons).
And remember .... when traveling (towing a trailer), or even a motor home ... you always have to build in "contingency time" in the event of mechanical failure. You may reach your destination, and then have 2 days to head back, pushing those 12-15 hour days. What happens if, on the first day returning home you have a breakdown, even a flat tire will delay you for hours. You need to have some "cushion time" in there somewhere. Pushing to the outer limits is risky.
Re-read all the comments on this thread and please consider what everyone is saying.