Is it permissible for me to apply for housing in the Mubarak National Housing project to secure a dwelling for myself, even though I am married and my husband owns an apartment? And would this be considered competing with more needy young people?

The Mubarak National Housing Project aims to provide 500,0 housing units for low-income individuals and needy youth, with the most deserving being those in greatest need. Based on your situation, it does not appear that you are in need of these apartments, and therefore, you should not compete with those in greater need. And Allah knows best.