How can one get rid of the feeling of doubt regarding the soundness of one's intention and faith and the verification of the truth of Islam, which afflicts the heart with despair and leads to the cessation of worship?

These are whisperings from Satan, which he casts to deter one from worship. Therefore, one must turn away from them and pay them no mind. The questioner's Islam is sound, and all such whisperings in this regard are from the devil's interference. So, one should not pay attention to them; rather, one must strive in acts of worship, have a good opinion of Allah, and not despair of His mercy.