Being under the new covenant means we are free from the animal and food offering we see outlined in the early books of the Bible. However they provide us with an understanding of how God wants to be approached in worship.
The various offerings for sin shows that God wants us to come to him with a clean heart free from sin which is now possible through Christ's death on the cross. However there were also other offerings such as the thank offerings which spring from the fellowship offerings that show a desire to be in fellowship with God.
Today, while it is not necessary to make the various food offerings to be in fellowship with God, it shows us the importance God places on having a thankful heart. When we have a thankful heart we have a truer perspective of our inadequacies and God's sufficiency. True thankfulness comes from humility, since we are thankful for those things which we cannot provide ourselves and we recognise God's goodness in his provision.
When we read Psalms we often find the phrase, "sacrifice thank offerings" which tells us thanksgiving wasn't an emotional response. Making these offerings were a sacrifice that was costly in time as well as money. It required preparation and commitment.
In the above verse we see David's commitment to fulfilling his vows and present thank offering to God. It wasn't dependant on whether he felt like it but on a decision he had already made.
This is quite a challenge. While we don't have to make the food preparations we still have to prepare our hearts and make a commitment to be thankful whether we feel like it or not.