I am more worried about those that are important to me than myself.
「私自身のことよりも大切な人のことが気がかり」は英語で「I am more worried about those that are important to me than myself.」という表現が良いと思います。
私自身ーMyself
大切な人ーImportant person/people・Loved ones・Significant other
気がかりーConcerned・Worried
「彼の健康が気がかりです。」I am concerned about his health.
「彼女がただ一つ気がかりなのは子供たちの安全だ。」Her only worry is the safety of her children.