Adamic men are the descendants of the mortal Adam. The ancient Jewish belief which I am describing here is that he was appointed 7,000 years in parallel to Creation week and his banishment - the last 1,000 years being the Sabbath, Christ's reign on earth. Thus concludes the "realistic, national" promise.
Last Adam:
When the 7,000 year calendar is over, this heaven and earth will be gone, replaced by the new heaven and the new earth populated by those of us whose names were written in the book of Life of the Lamb from the foundation of the world. Thus concludes the "transcending, universal" promise, the preexistent Messiah, Jesus Christ by Whom and for Whom "all that there is" is.
It is not an either/or - not for the Jews (who cannot see the later) - and not for us (who sometimes cannot see the former.)
I know that the Jewish belief is "in the world to come" and that world is a physical reality. I don't know if it's also spiritual, but I would guess so.
However, everything that they believed has been clarified in Christ, whether they believe it or not, so I can't buy an Adamic rest for anyone unspiritual.
The meek will inherit the earth -- even that would imply spirituality, as both Moses and Christ are said to be meek