"Anyway I do not need him."
Translation:Dù sao tôi cũng không cần anh ấy.
Yes, it's correct. But the structure "dù sao Subject cũng/vẫn" make more sense.
I think with "anyway" = "dù sao", it would be natural to have "cũng" because you're reaffirming your statement. You probably already stated that you don't need him. So anyway (to reaffirm), I still (cũng) don't need him.
How does the word "cũng" make more sense? Thank you for explaining in more detail :-)