Susan,

I've not had much time available to respond to your important question so I'm thankful Pilgrim has stepped in on my behalf and explained the broader view of the Covenant of Grace which paedobaptist's embrace.

There's an excellent article by Prof. John Murray here on the-highway which sheds more light on the The Covenant of Grace. In this article he addresses the importance of God's covenant with Noah, Moses, Abraham and David. Specifically the Covenant of Grace made with Abraham. He also discusses keeping covenant, breaking covenant, and how the old is fulfilled in the new. This study may answer many questions you may have.

Quote
Murray writes:

When we come to those passages in the New Testament which deal specifically with the new covenant in contrast with the old it is highly significant that the contrast between the new economy and the old is not expressed in terms of difference between covenant and something else not a covenant. The contrast is within the ambit of covenant. This would lead us to expect that the basic idea of covenant which we find in the Old Testament is carried over into the New. We are confirmed in this expectation when we take account of the fact that the new covenant is the fulfilment of the covenant made with Abraham (Lk. i. 72; Gal. iii. 15ff.). The new economy as covenant attaches itself to the Old Testament covenant promise and cannot be contrasted with Old Testament covenant in respect of that which constitutes the essence of covenant grace and promise. We can express the fact that the new covenant is the expansion and fulfilment of the Abrahamic by saying that it was just because the promise to Abraham had the bonded and oath-bound character of a covenant that its realization in the fulness of the time was inviolably certain. The new covenant in respect of its being a covenant does not differ from the Abrahamic as a sovereign administration of grace, divine in its inception, establishment, confirmation, and fulfilment. The most conclusive evidence, however, is derived from a study of the New Testament respecting the nature of the new covenant. We shall find that the features of the covenant are the same as those we found in connection with covenant in the Old Testament.

When our Lord said that His blood was the blood of the covenant that was shed for many for the remission of sins and that the cup of the last supper was the new covenant in His blood (Mt. xxvi. 28; Mk. xiv. 24; Lk. xxii. 20; 1 Cor. xi. 25), we cannot but regard the covenant as a designation of the sum-total of grace, blessing, truth, and relationship comprised in that redemption which His blood has secured. Covenant must refer to the bestowment and the relationship secured by the sacrificial blood which He shed. It is the fulness of grace purchased by His blood and conveyed by it. By way of comparison there is an allusion, no doubt, to the blood by which the old covenant, the Mosaic, had been sealed (Ex. xxiv. 6-8; cf. Heb. ix. 18). And since the new is contrasted with the old it cannot be that the contrast inheres in any retraction or dilution of the grace which we have found to be the essence of covenant under the Old Testament.


Wes


When I survey the wondrous cross on which the Prince of Glory died, my richest gain I count but loss and pour contempt on all my pride. - Isaac Watts