I wanted to ask you, is taking care of ones ‘sexual’ desires (in a halal way) mean that one is running after this world?

No it doesn’t mean one is running after this world. The Prophets and Imams would also address this need they had in a halal way.

However, if it becomes an obsession, or one is willing to commit haram or abandon one’s obligations for the sake of sexual desires, then this would be running after this world.