The Law of Moses is a burden. Grace doesn't make the law any less of a burden. Life in Christ doesn't make the law any less of a burden. Peter addressed this in Acts 15 when some Pharisees said it was necessary for Gentile believers to keep the law.
First, note what Peter didn't say. He didn't say, "Well, now that they have Christ, they indeed are able to and should keep the law." To the contrary, he said to those Pharisees, "Why do you test God by putting a putting a yoke around the neck of the disciples which neither our fathers nor we were able to bear?"
For centuries the law had been a burdensome yoke for the Jews, and it doesn't become any less of a burden for those who believe in Christ who try to live by it. The law was always meant to be a burden. Its purpose was to point people away from self-righteous justification to justification by faith in Christ alone.
In Christ, God has removed the burden. He hasn't done this by making the law easier to keep, by grace. Rather, He's "wiped it out" and "taken it out of the way, having nailed it to the cross." (Col 2:14) So why are so many believers still living under it? Let's not add to their burden by teaching and preaching law in the lives of believers!
As we saw in Acts, Peter had referred to the law as a burdensome yoke that no one could bear. Paul took this a step further in his word to the "foolish Galatians" who knew they'd been saved apart from law but had gone back to the law as a means of attempting to complete the work that was begun by the Spirit and could only be accomplished by grace. Paul exhorted them, "It is for freedom that Christ set us free. Stand firm then, and do not let yourselves be burdened again by a yoke of bondage." (Gal 5:1)
The law is, was and always will be a yoke of bondage and a burden. What we need is not more or better adherence to the law, but rather to be free from it entirely! When we're freed from the law, we're freed into the liberty of Christ. We're freed from death unto life. (2 Cor 3:7-9, Rom 5:17, Rom 8:2). We're freed from sin unto righteousness. (Rom 6:7, 18-22). This is what Christ has done for us!
The "Spirit of life in Christ Jesus" is our life now! This is where our hope is. This is our source of righteousness, and in fact is our very righteousness. This is where our godly living comes from. Put the law away and turn to Christ!