“they will bear their shame and the consequences of the abominations that they have committed.” Many people who are saved and have everlasting life never make the effort to live a truly godly life on earth. In the future kingdom of Christ there will be a distinct difference between those people who made the effort to live truly godly lives and those people who got saved but continued living selfishly and in sin. The people who have little or no rewards in the Millennial Kingdom will realize why that is, and will be ashamed of their selfish lives. In this case in Ezekiel 44:13, there will be priests who did enough to get saved, but worshipped both God and idols, and they will bear the consequence of their idolatry.
[For more on shame in the future life, see commentary on 1 John 2:28 and 2 Cor. 5:10, “good or worthless.” For more on the future Millennial Kingdom, see Appendix 3, “Christ’s Future Kingdom on Earth”].