You could take an amtrack train if you really wanted to, couldn't you?
You could also rent a car and drive. With one overnight stay, it isn't too long a trip.
Also, realistically, many, many, many people are flying without incident, I don't know if your friends are giving you good advice. But, it is your vacation, and being bothered about the arriving and returning flight may not help with the relaxation you deserve. The two options above may help.
I have very fond memories of train trips when I was young with my parents, it is more expensive, but the trip starts when you leave the station, and you can have your own stateroom.
Amtrack has a pretty good website to help with fares and availability. There is also an autotrain program that sounded pretty neat and might be fun to try sometime. We drive by the Amtrack autotrain sign on I-95 to and from Vero Beach.
Best wishes and I hope your vacation is wonderful.