不安 and 安心: following changes in worry
Track the content of a worry, what brings relief, and anxiety that remains over time.
What the worry concerns
不安, read ふあん, describes an uneasy or worried state. 明日の面接が不安です。 means that someone is worried about tomorrow's interview.
In うまく話せるか不安です, the worry concerns whether the person can speak well. Do not turn the clause before か into a failure that has already happened.
What brings relief
安心, read あんしん, describes relief as worry subsides. 先生の説明を聞いて、少し安心しました。 says that hearing the teacher's explanation brought a little relief.
Keep the limited degree expressed by 少し. Do not expand this into the complete disappearance of every worry.
Worry that remains
説明を聞きましたが、まだ不安です。 says that the person heard the explanation but is still worried. The event contrasts with the feeling afterward.
不安がなくなったら、申し込みます。 says that the person will apply once the worry is gone. Applying is a plan conditional on that change, not a completed action.
Can 少し安心しました be translated as every worry disappearing?
Show the answer and why
No. It expresses a little relief, leaving open the possibility that some worry remains.