What is the ruling regarding someone whose term of life expired, and they had intended to embrace Islam and declared it publicly? And will their sincere intention intercede for them?

Entering Islam is not determined by mere intention. Rather, a person must pronounce the two testimonies (Shahadatayn) if they are capable of uttering them. The People of the Sunnah have agreed that a believer, who is judged as such, is one who believes in the religion of Islam in their heart with firm conviction, free from doubts, and pronounces the two testimonies. Whoever does not pronounce the two testimonies is an infidel outwardly and inwardly, and the rulings applicable to infidels apply to them.