When you're in sand the additional load forces the transmission's torque converter to "churn" the fluid which typically generates considerable amounts of heat and that would have transferred to the vehicle cooling system. Excessive heat will shorten the life of the transmission fluid and also the transmission, you may want to consider changing the fluid more often.
If you're going to be in soft sand frequently, consider fitting an external transmission cooler and a transmission temperature gauge, fit the temperature gauge sender on the output line from the transmission to the external cooler, and then go from the external cooler to the heat exchanger in the bottom of the radiator, and then back to the transmission as normal. The reason I'm specifying how you sequence these items is that with the gauge sender at the transmission output you know how hot the fluid is getting, the external cooler will "dump" the heat before it reaches the cooling system, and in the event that the transmission fluid becomes "over cooled" by the external cooler, it will be warmed by the heat exchange in the radiator - this last is only a requirement in climates that experience colder weather.
I see no need to replace the radiator, unless there is an issue with it - at this point it appears that the heat you're concerned about is a "one off" instance in the outer banks, so I'm thinking it was caused by the sand, rather than a cooling system fault. The timing belt is a service item, and needs to be done, I believe at 60,000 mile intervals - if you don't know the change history, it's a good idea to get it done, many folks replace the water pump at the same time because of the labor costs that would be required if it had to be done separately, however, I personally see no reason to do that, possibly because I can do it myself in my driveway and don't have to pay a shop.
A cooling system flush & refill with fresh coolant is also a service item, so if you don't know when last it was done, it's a good idea to do it, along with all the other fluids.