Sefer Hatteruma addresses this issue (closely imitated in Hilchot Erets Yisra'el attributed to the Tur). They bring a few examples that prove that the ashes of the red cow were still in use after the destruction of the Temple, until well into the days of the amora'im, even though it must have been made while the Temple still stood.
Nidda 32a: Rabbi Yosi, Rabbi and Rav Yosef all mention cases of immersing a newborn girl who became nidda, so that she could touch teruma or be anointed with teruma.
Chullin 107b: Rabbi Ami and Rabbi Asi ate teruma with wineskins so as not to contaminate the teruma with their hands. If there were no ashes of the red cow, they would have been impure in other ways and wouldn't have been concerned only with the impurity of their hands.
Nidda 6b: In the days of Rabbi, there were still people who were careful to keep their food pure to be used by the Temple when rebuilt.
Bechorot 27b: Rav Nachman asks rhetorically "Do we have the sprinkling of the red cow?" meaning that in Babylon they didn't have it, but in Israel they still had it (Tosafot there seems to have assumed the same).
Interestingly, none of these cases actually says anything about the ashes of the red cow. The Sefer Hatteruma deduces that it must have existed for legal reasons. Otherwise, these aren't exactly direct proofs. Since I assume he would have quoted any explicit statements if there were any rather than relying on the implication of these passages, there are probably no more explicit statements about the amora'im using the ashes of the red cow.
In summary, I think that it's safe to say that many people in the times of the early amora'im they were still careful about purity of their food, and that some rishonim took this as proof that the ashes of the red heifer were still in use at that time.