The trailer you own is big, if you really want an F150, you REALLY need the HD Payload package as well as max tow. Max tow by itself will not get you the payload you need to pile the family in, stuff in the back, and tow the trailer. The HD Payload package is easily determined as it will have 8200lb GVWR and 7 lug axles. It looks like HD Payload can be had with the ecoboost or 5.0 but not the 6.2 according to the Ford specification page. Even well equipped the HD payload package will still give you north of 2000lbs payload. Yes, they exist but many salesmen don't know about them. Unfortunately, you are not likely to just find one on a lot, you will probably have to order it.
The sticker you posted has a 7200lb GVWR which means no max tow, and the other one with 7700lb GVWR but only 1136lbs may have max tow but so may gizmos added that it has been reduced to a grocery getter. After taking family and basic stuff out of that rating you are left with around 500lbs for the trailer tongue.
Anything short of a HD F150 or F250 is not going to be an improvement over your current configuration.
