In Ultimate Vault Hunter Mode, you must use the right elemental weapons.
Shock against shields (blue bar)
Fire against flesh (red bar)--> note some fleshy enemies are resistant to fire, like incenerators and fire skags.
Acid against armored enemies( yellow bar)
you can also use slag weapons to apply slag effect on enemies, then switch to another weapon for x2 damage.
In borderlands 2, there is no need to farm xp, simply doing all the side quest yeld enough xp to stay ahead in levels.
Also remember to upgrade your gear constantly, even legendaries become worthless in 5-6 levels, so it is important to have your gear to match your current level, even white items can sometimes be better than few levels lower green and blue items.
Regarding the quest you are stuck in, keep your distance, second assassin has very poor accuracy, but strong attack power. By constantly kiting him, you can avoid almost all damage completely, just stay on the move to avoid his explosive attacks, destroy the shields with shock weapon, apply slag and then use fire to burn him alive.