In Titus 2:11-14, our salvation is described as being trained by grace to do what is godly, righteous, and good, and to renounce doing what is ungodly, which is what the Mosaic Law was given to instruct how to do, so God graciously teaching us to obey it is itself part of the content of God's gift of salvation, and participating in that training does nothing to earn it, but rather that is how we receive it. In Titus 2:14, Jesus gave himself to redeem us from all lawlessness and to purify for himself a people of his own possession who are zealous for doing good works, so becoming zealous for doing good works in obedience to the Mosaic Law is what it means to believe in what Jesus through his resurrection (Acts 21:20). Our salvation is from sin (Matthew 1:21) and sin is the transgression of Mosaic Law (1 John 3:4), so being trained by grace to live in obedience to God's law through faith is what it means to receive the gift of Jesus saving us from living in transgression of the Mosaic Law.
Jesus was sinless, so he would have still taught how to walk in complete obedience to the Mosaic Law by example even if he had said nothing, and as his followers we are told to follow his example (1 Peter 2:21-22) and that those who are in Christ are obligated to walk in the same way he walked (1 John 2:4). Jesus did not hypocritically preach something other than what he practiced and he did not establish the New Covenant in order to undermine anything that he spent his ministry teaching, but rather the New Covenant still involves following the same law (Hebrews 8:10). In John 12:44-50, Jesus did not give his followers any room to disregard anything that he taught during his ministry. In John 14:24, Jesus said that his teachings were not his own, but that of the Father, so he taught the same commandments as the Father. In John 15:10, Jesus used a parallel statement to equate his commandments with those of the Father. In Deuteronomy 4:2, it is a sin to add to or subtract from the Mosaic Law, so Jesus did not give his own commandments in disagreement with what the Father has commanded, but rather everything that he taught was in accordance with the OT, and in fact there has never been someone else whose teachings were more thoroughly rooted in the OT.
In regard to John 13:34, there was nothing brand new about the command to love our neighbor because it can be found in Leviticus 19:18, but what was new about Christ's command was the quality of the example by which we should love our neighbor, and indeed the Greek word used refers to newness with respect to quality rather than with respect to time:
3501 /néos ("new on the scene") suggests something "new in time" – in contrast to its near-synonym (2537 /kainós, "new in quality").
We should love ourselves as God loves us and that is how we should love our neighbor, so Jesus was not sinning in violation of Deuteronomy 4:2 by making changes to the Mosaic Law, but rather he was fulfilling the Mosaic Law by teaching how to correctly understand and obey it.
Obedience to the Mosaic Law can be done for purposes other than earning our salvation, especially because it was never given as a means of doing that. Even if someone managed to live in perfect obedience to the Mosaic Law, then they still would not earn their salvation because it is not something that can be earned (Romans 4:4-5), so that has always been a fundamental misunderstanding of the goal of obeying it. In Jeremiah 6:16-19 and Matthew 11:28-30, the Mosaic Law is described as the good way where we will find rest for our souls, but thinking that our obedience is about needing to have a good enough job would rob our souls of the rest that it is intended to give.