So would it be appropriate to assume that the 'administration' of the covenant of grace in the OT is functionally the same thing as the 'dispensation' of the Law within that same covenant?
WCF 7, Of God's Covenant with Man
7:5 This covenant was differently administered in the time of the law, and in the time of the gospel: under the law, it was administered by promises, prophecies, sacrifices, circumcision, the paschal lamb, and other types and ordinances delivered to the people of the Jews, all fore-signifying Christ to come...
7:6 Under the gospel, when Christ, the substance, was exhibited,... There are not therefore two covenants of grace, differing in substance, but one and the same, under various dispensations.
In para.5, the phrase "time of the law" is used in a broad sense, basically covering the ages prior to the coming in of Christ. The original covenant of works looms over that time, and law threatens all because all sinned. We are made aware that there is another principle coming forth, one of grace, which is also operative in that "time" but awaits full revelation in the Person of Christ, and the "time of the gospel."
The phrase "under the law" then takes in the whole OT by inclusively speaking not only of promises, but also the primary exhibition of that age in the Mosaic or Siniatic administration, or Law of Moses.
"Administration' of the covenant of grace in the OT is not functionally the same thing as the 'dispensation' of the Law, for this reason: there was more than one administration of the CoG in the OT.
There is a "proto-administration" from Adam post-fall until Abraham. The CoG is operative starting with Gen.3:15 (if we ignore the Son's appointment from before the foundation of the world); but there is little in the way of recorded revelation directly to its point. Sin is the problem, death is the too obvious result, but hope has been promised.
Starting with Abraham, we have new and (in contrast to before) brilliant revelation. The promise is given fresh life, and a vector, and a plain covenant expression. It is the one Seed of Abraham that will provide the solution to the problem; and to bring it about there are intermediate promises that will be fulfilled step-by-step on the way. And the end result is not something that this world can contain. So Hebrews tells us that Abraham was not looking for his own earthly fulfillment (a city).
The "dispensation/adminstration of the Law" is strictly speaking the time of Israel under the Law of Moses. There are several "phases" of this administration, which if one knows OT history he can find the contours. In general they are: in the wilderness, in the land, under the kings, exile/return. In the center of this administration you can find the climactic promises of the CoG made to David.
Hope this is helpful. Of course, much much more can be said.
