Apart from the conditional mosaic covenant all covenants are everlasting in one sense.I think that we are looking at different "everlasting covenants". When I hear the term, I think of God's everlasting covenant between Himself (the Trinity) and His people. Throughout Scripture the people of God appeal to this covenant specifically because it was made with them. If you will, please provide a few passages so that I can better see the distinction between this and the everlasting covenant of which you are referring.
Here is the one I'm talking about (are we looking at the same thing differently, or is there another everlasting covenant?).
Genesis 17 "I will make you exceedingly fruitful, and I will make nations of you, and kings will come forth from you. I will establish My covenant between Me and you and your descendants after you throughout their generations for an everlasting covenant, to be God to you and to your descendants after you...
Jeremiah 32 "They shall be My people, and I will be their God; and I will give them one heart and one way, that they may fear Me always, for their own good and for the good of their children after them. I will make an everlasting covenant with them that I will not turn away from them, to do them good; and I will put the fear of Me in their hearts so that they will not turn away from Me. I will rejoice over them to do them good and will faithfully plant them in this land with all My heart and with all My soul."
Ezekiel 16 "Nevertheless, I will remember My covenant with you in the days of your youth, and I will establish an everlasting covenant with you. Then you will remember your ways and be ashamed when you receive your sisters, both your older and your younger; and I will give them to you as daughters, but not because of your covenant. Thus I will establish My covenant with you, and you shall know that I am the LORD, so that you may remember and be ashamed and never open your mouth anymore because of your humiliation, when I have forgiven you for all that you have done," the Lord GOD declares.
Ezekiel 37 "I will make a covenant of peace with them; it will be an everlasting covenant with them. And I will place them and multiply them, and will set My sanctuary in their midst forever. My dwelling place also will be with them; and I will be their God, and they will be My people. And the nations will know that I am the LORD who sanctifies Israel, when My sanctuary is in their midst forever."
For the sake of this discussion the arrangement between the 3 persons of the trinity commonly known as the Covenant of Redemption stands for all eternity even before man was actually created. Jesus as the mediator and surety for the elect render it certain....the elect benefit from this going forward. I would examine jn 6, jn10, and jn17 here.
The excellent verses you offered are a part of the progressive revelation of this covenant to man, and I think that this was what Biblicist was alluding to.