Concerning Hebrews 8:13...
It is in context of Hebrews 8:8-10, in which it is written that the law is written on the hearts and in the minds of all those who are under the New Covenant. See also Hebrews 10:16.
So then, it is the ceremonial law that is growing old and is ready to vanish away...because the Lord fulfilled it in dying on the Cross.
But the rest of it...the moral tenets, statutes, and precepts...these are written on the heart and in the mind of every New Covenant believer.
For this is the love of God, that we keep His commandments (Romans 13:8-10, 1 John 5:3, 2 John 1:6).
And the love of God comes into our hearts by faith (Galatians 3:14, Romans 5:5).
So, the New Covenant is indeed better; for it provides the means by which we can become righteous; whereas the Old Covenant was never able to do that.
However, it should be clear that the Old Covenant is not nullified by the New when it comes down to defining righteousness and sin; for the law defines sin (Romans 3:20, 1 John 3:4) and righteousness.
One jot or one tittle shall in no wise pass away from the law until all is fulfilled...and among those things that must be fulfilled is that heaven and earth must pass away.
Therefore the change in law spoken of in Hebrews 7:12 cannot be speaking of a removal of certain moral tenets found in the law.
Now, not only is the law written on the hearts and in the minds of New Covenant believers; but the law serves as a schoolmaster to lead men to Christ (Galatians 3:24, Romans 3:20, Psalms 19:7 (kjv))
Psa 19:7, The law of the LORD is perfect, converting the soul: the testimony of the LORD is sure, making wise the simple.
The law shows man his sin; that he is a sinner in need of the Saviour...because it is written that the Saviour shall save His people from their sins (Matthew 1:21) and therefore the law will convict you for as long as you continue to be a sinner.
When you repent and give your heart to the Lord, you cease to be a sinner (1 John 3:9) because you have made a 180-degree turn away from sin and towards righteousness.
This is not the righteousness which is of the law; it is the righteousness which is of God by faith.
Nevertheless, those who have faith are given the Holy Ghost (Galatians 3:14)...and as for those who walk according to the Spirit, the righteousness of the law is fulfilled in them (Romans 8:4).
So the righteousness which is of God by faith produces the righteousness which is according to the law...a righteousness which is without the law but is nevertheless attested to by the law and the prophets (Romans 3:21).
Because the fruit of the Spirit is love and goodness, among other things...and against such there is no law (Galatians 5:22-23).
Now as concerning tithing, I will say again that the OT gives a promise that the windows of heaven shall be opened to the nation that brings the whole tithe into the storehouse...and that this is the promise of the Lord.
For Jesus Christ is the same yesterday, today, and for ever (Hebrews 13:8, Malachi 3:6).