Will a person be held accountable for feeling disgusted towards someone they see committing a sin, knowing that everyone sins, and is it always necessary to offer advice, even if it is not accepted or if one fears embarrassment?

What you find within yourself of hatred for sins and their perpetrators is praiseworthy and incurs no sin. One should bear in mind the benefits of hating disobedient individuals seeking the countenance of Allah. The believer hates the disobedient person for the sake of Allah to the extent of their disobedience, based on the Prophet's (peace be upon him) saying: "Whoever loves for Allah, and hates for Allah, and gives for Allah, and withholds for Allah, has perfected faith."

Along with this, it is necessary to be keen on enjoining good and forbidding evil to the best of one's ability.

Sheikh Ibn Baz clarified that boycotting an innovator (mubtadi') or a disobedient person (a'si) is a right or a Sunnah if it does not lead to a greater evil. However, if not boycotting them is more conducive to inviting and guiding them, then one should not hasten to boycott. Rather, they should be hated for the sake of Allah to the extent of their innovation or disobedience, and loved for the sake of Allah to the extent of their Islam and faith.