Is it a prerequisite for spouses to separate, and is it permissible to touch the Qur'an and pray after performing ghusl and uttering the Shahada, in order to be rid of the feeling of objecting to God's decrees?

Your hatred for the whispers (waswasa) that call you to object to Allah's rulings is a sign of good and an indication of pure faith, because a believer dislikes what contradicts faith, as narrated from the Prophet, peace and blessings be upon him. Al-Nawawi and Ibn Mandah mentioned that considering these thoughts to be grave and fearing them is pure faith itself. Your faith is not affected by these whispers as long as you hate them and strive to repel them. Your relationship with your husband is also not affected by this, so beware lest Satan makes you abstain from him. For treatment, increase your prayers and supplications, completely ignore the whispers, and dedicate yourself to learning religious matters, for knowledge protects against Satan's whispers, while ignorance makes you vulnerable to them. The intended doubt that affects faith is doubt regarding whether Sharia rulings are from Allah or whether they are purely good for the servants.