Yes, the legendary drop rate is better while you are on the mission. The warrior does drop other orange items but the legendary drop rate after completing the mission is much lower.
Other users talk about using the moonshot method to farm for an orange item while still holding onto the mission. I do not know the specifics though.
Even though the warrior's legendary drop rates are low, I farmed him for while and got some outstanding purple items that I have yet to replace. --- Fact because it's science. ==============================================
For what it's worth, my understanding, which is likely lacking, of the Moonshot method goes like this:
Beat The Warrior. Call the Moonshot. Got what you wanted? No? Dashboard out of the game. Load back up, you should still be at the 'Call the Moonshot' part. If so, Call it and hope you get what you want,
You might have to Save while at the Call the Moonshot part to stay at that when you Dashboard out, but otherwise, that's the trick.
Moonshot makes Warrior drop loot, and while on Talon of God, the Warrior has a relatively massive drop rate for Orange tier loot, thus, repeating the Moonshot over and over repeats the chance for Orange stuff. You might even get 2+ oranges on the same run! No idea if you keep anything if you Dashboard though :x --- Spell card! Diamond Will ~ Unbreakable Determination!