First, where did you get the numbers? If it is from the internet or some published source, forget them. they are wrong!! My '13 F350 diesel has only 3453# payload. The diesel adds 800#, but the F250 loses 1500# of GVWR.
If the truck is 2009 or newer, there is a yellow label on the driver door B pillar that states the max carrying capacity for THAT truck as it left the factory. Use this number to determine max FW capacity.
Add the weight of all passengers, pets, tools and cargo to be carried in the truck. Add 200# for a FW hitch. Subtract this from the Payload (max carrying capacity) determined above. This is the maximum pin wt the truck can carry. Since pin wt averages about 20%, multiply by 5 to get the max FW LOADED FW weight you can tow.
You can use that 14,000# number, but you must subtract the weight of all options in the truck and the weight of all passengers (except 150# for driver), cargo, and even the FW hitch from this weight.
You CAN tow a FW with a F250 gasser, but 14,000# may be a bit optimistic. Remember, that is the loaded weight, NOT the dry weight the sales staff loves to quote - unless you plan to go camping with an empty camper!
As Outdoor said, (once the weight question is settled), the most important factor is FLOOR PLAN. It's not what I like, but what will work best for you and your family. Go shopping with the check book at home. Sit in as many as your knees will let you get into. Some floor plans and options will demand your attention, others will scream NO! You learn this from looking.
Shop and research. Enjoy the journey and be safe!