Yes, an F250 will be better.
You will still need to pay attention to how loads are located, both in the TT and in the truck. With a heavier payload available with the bigger truck (assumed), you'll be able to carry cargo in the bed or maybe even shift some of the loads you have in the trailer to the truck.
I'd guess your tongue weight on the truck is approaching the limit of the receiver even with a WDH. That tongue weight is what is making your travels in windy conditions a nightmare. The 3.31 rear end gearing is what's making hill climbs slow and plodding.