Is it obligatory for me to pray in the mosque standing as much as I can, then sit if the Imam prolongs the prayer, or should I complete the prayer at home standing with a short prayer, knowing that I am young and feel embarrassed to pray sitting?

If you are able to pray standing individually, but cannot do so in congregation except by sitting for part of the prayer, then it is better for you to pray standing alone. Your prayer with the congregation is valid. This is the view of the Shafi‘i school, and it was explicitly stated by Al-Khatib Al-Shirbini. As for the Hanbalis, they offered a choice between the two options, and some of them preferred praying individually while standing, because standing is a pillar that does not fall except due to an excuse, whereas congregational prayer is an obligation for which the prayer is valid without it.