No, the paladin can not resist the damage.
The Elemental Adept feat says:
When you gain this feat, choose one of the following damage types: acid, cold, fire, lightning, or thunder.
- Spells you cast ignore resistance to damage of the chosen type. [...]
While the Warding Aura of the paladin says:
Beginning at 7th level, ancient magic lies so heavily upon you that it forms an eldritch ward. You and friendly creatures within 10 feet of you have resistance to damage from spells.
In this case, the spell damage consists in fire damage: the spell description says (emphasis mine)
[...] A target takes 8d6 fire damage on a failed save, or half as much damage on a successful one.
The Warding Aura grants resistance to damage from spells, in this case the resistance is granted against the fire damage caused by the spell Fireball. But there is no particular rule that attests that the paladin's Warding Aura overcomes the Elemental Adept feat: hence the paladin (and their friendly creatures within 10ft of them) can not benefit from resistance to fire against the Fireball cast by the Draconic Sorcerer.