No they didn't. Korra lost her past lives... which was just their knowledge and power. The avatar state has nothing to do with her past lives, but the light Spirit Raava. The entire point of "The Beginnings" was to explain all of this. The avatar state comes from Raava, the past lives merely add their knowledge and power to the avatar. Raava is still there, therefor the avatar state still exists. Otherwise how would Wan be able to enter the avatar state, considering the fact he was the one and only avatar(thus no past lives).

I remember reading about the finale of Season 3 and someone mentioned the comparison of Korra's avatar state vs Aang's. They noted that Aang's avatar state was extremely powerful and tactical. Aang was able to use so many high-level techniques(such as the boulder machine guns), which require knowledge of said techniques(like metal bending)... on the other hand Korra's state lacked any kind of finesse, and instead was just raw power. The point is that when Aang entered the avatar state all the knowledge of the past lives contributed their skill and technique, whereas with Korra she had no past lives... thus no knowledge of advanced techniques. Instead since it was just Raava, all she got was a massive boost in bending power.