When someone goes up to the bima for a bracha on the Torah I get up and I say my part of the bracha, and at the end of the bracha I sit down to listen to the reading of the Torah. Is this correct?

There are many acceptable customs. You are following the psak of the Mishna Berura (146 18) who requires one to stand when "barchu" is said by the oleh and when one responds with "baruch... hamevorach l'olam vaed". There are those who stand for the entire Torah reading although this is not required. Those who sit even during "barchu" and its response rely on rabbinical sources that don't require one to stand at all (Ishei Yisrael 38 19 n.51--53).