Korach had to die an unnatural death; otherwise his death wouldn't prove Moshe right, because a natural death could be because his time had come. But why did he have to still stay alive after being buried which was something new and had never been before or since? The wording of the Torah is 'chaim sheolo'. Meaning alive in the grave. If it just meant being buried alive and dying later there would be no point in the chumash mentioning it. In all earthquakes which this was people are buried alive and die later.
"They went down to the pit alive" - even in the grave they think they are alive.
There is a blessing contained in "They went down to the pit alive," as with "the sons of Korach did not die," - "a place was established for them and they repented." For teshuva, repentance, is effective only while one is still alive. This, then, is the blessing - that even in the pit they will live, and they will be able to effect teshuva.
According to the first paragraph, then, they may not indeed have been alive after their descent - but they were so spiritually degraded that they didn't realize that they had been discredited. According to the second paragraph - on the contrary, it means that there is still hope for them. (Possibly, these two explanations reflect the argument about whether Korach's group lost their place in olam haba (R. Akiva in Sanhedrin 108a and 109b) or not (R. Eliezer in 108a, R. Yehudah ben Beseira in 109b).)
The pesukim in question (Bemidbar 16:30, 33) are most explicitly about Datan and Aviram, not necessarily about Korach. He might have been burned by the fire, or even both.
But the question you asked could be asked just as well about Datan and Aviram.
Sheol is the grave. And perhaps also the underworld in Biblical Hebrew at this stage.
But the simple peshat in the pasuk is not that they stayed alive after being buried. Most people first die, and then are placed in grave. Here, these people went into the grave alive, as the earth swallowed them. Thus, וְיָרְדוּ חַיִּים שְׁאֹלָה, "and they descended alive into the grave. During their descent, they were still alive. That does not mean that they did not immediately asphyxiate.
Is this your own interpretation, from which you are asking? Or it this a known midrash?
There is a midrash about the sons of Korach staying alive for quite some time after, but that is an entirely different idea.
I read in the name of the kabalists (could be arizal) that Korach was a gilgul of Kain and Moshe was a gilgul of Hevel. Since, the "earth swallowed the blood of Hevel" as written: "And now, you are cursed even more than the ground, which opened its mouth to take your brother's blood from your hand" (Bereishis 4:11)
Therefore justice demanded that Korach be swallowed and suffer to atone for what he did.