Which Should I have? I have 3 slots, a BlazeBrand, and a Kilowatt pulsar, I have Ash Tail armor, and a 4* Owl shield. Which would be better for me? Also, I saw avenger has elemental. What element?
Faust or Avenger?
Optimally, you want an elemental, shadow, and pierce sword (or gun, or bomb depending on what your main weapons are) so you can deal optimal damage to each type of monster. You can go to the wiki if you don't know which monsters are resistant to which attack types.
you already have an elemental sword, so if it is just a tossup of those two swords, then the Faust would be more useful for you now. Once you have a weapon of each type, you may want to expand your arsenal to include various weapons that have different charge attacks or status inflicts.
My advice on these swords is here. The other posters are right, that DA would be redundant with your other elemental sword.
"Also, I saw avenger has elemental. What element?"
Perhaps you are confusing elemental damage with status effects. DA doesn't do any status. Just elemental (and normal) damage.
If you are a newb who just got 20 jelly gems, always get the Faust first. It's a shadow sword, so it'll make killing Royal Jelly easier (and help you get your Avenger faster).
Otherwise it comes down to what swords you already have in your arsenal. In your situation, you lack a shadow sword…so still get Faust.
Faust, because you already have an Elemental/Normal sword
Faust, since you can use that to farm jellies faster and thus get another sealed sword to upgrade into avenger.
Elemental itself is a type of damage. There is no fire element, water, wind earth. Nothing like that in SK. If you were thinking of status, it does not have one.