The first book you describe might be Star Rangers, by Andre Norton. (The link will take you to its Wikipedia article.) I have fond memories of when I first encountered that book in a library when I was a young whippersnapper. And I've occasionally reread it (although I suspect it's been over a decade since I last did).
But if I'm right, then I think your memory has blurred some of the nitpicking details. (The same thing has happened to me several times when I come here to ask for help in identifying an old story.)
Let me cover some of the points which make me think this is what you were describing:
You said:
This means they were originally published in the late 1970s at the
latest, but I would guess they were older than that.
According to the Wikipedia entry, Star Rangers was first published in 1953. It has also been called The Last Planet (something I didn't know until just now).
You said:
Both take place in a galaxy-spanning empire that is in decline; this
decline is apparently irreversible, and has been going on for a very
long time, on the order of centuries or millenia, as the central
government's resources diminish and as there are rebellions on the
peripheries. Technology is advanced, but stagnant. The central
government is referred to as "Central Command", often abbreviated to
CC.
Wikipedia says:
The First Galactic Empire is disintegrating and petty tyrants are
creating their own fiefdoms. Near the Empire's edge, a Central Control
agent seeks to rid himself of the Stellar Patrol, the last protector
of law and order in the empire, by sending it out to locate lost
stellar systems. Ships are sent to retrieve and align these systems
under the benign rule of Central Control. The ship at the heart of the
story is the Scout ship, "Starfire".
"Central Control" is not far removed from "Central Command."
You said:
Mutants, and "greenies", are discriminated against, or worse.
Wikipedia says:
Living among the humans are several kinds of aliens. For example,
there are the "Bemmies" whose name comes from BEM (short for Bug-Eyed
Monster)
So if you got "greenie" confused with "Bemmie," this would be a good fit.
You said:
In one book, the story centers on the crew of a military vessel, which
crash-lands on an uninhabited but pleasant world, which turns out to
be Earth.
Wikipedia says:
In the year 8054 AD, the Stellar Patrol Scoutship Starfire has
crashed in a desert on an Earth-like planet. The planet's atmosphere,
gravity, and solar radiation are almost ideal for the Rangers and the
Patrolmen. On initial examination, there are no signs of civilization.
After burying their dead, the survivors set up a camp in a forest
beside a river.
I can verify from personal recollection that it is only in the final pages of the book that the main viewpoint character (and his friends) realize that they have found "Terra of Sol" -- the legendary birthplace of humanity, now very sparsely inhabited (but with one or more old cities, empty, which show this world used to be much more densely populated and very high-tech).
You said:
At the end of the novel, the protagonists meet the crew of another
ship that had crash-landed on Earth; they decide together to turn
their back on Central Command and settle on Earth.
Wikipedia says:
As they marvel at the discovery, a band of refugees from a Stellar
Patrol base which was destroyed by pirates joins them. As ranking
officer, Kartr asks for a vote on whether to return to the city or try
to live in the wilderness. The people vote unanimously for a new start
and Kartr leads them into their future.
I might mention, though, that a bunch of refugees from at least one other ship were also involved in the plot of this novel. In other words, within a period of, let's say, a few weeks, at least three different ships all made landings (usually emergency landings, I think) on the same long-forgotten planet . . . by sheer coincidence, I believe!
Regarding the other book which you believe is linked to this one as part of the same series, you said:
The other book is also set on Earth; it seems to take place sometime
in the near future. I don't know how it relates in a timeline to the
other novel I described, whether long before or long after. At some
point, there was a reference to the city of Sacramento, as an
agricultural center. Earth is considered to be under the oppressive
authority of Central Command, but is a "barbarian" world. The
protagonist, early in the novel, sees a sunlit wall, and thinks he'd
like to stretch out on it, but does not, because that would identify
him as a "greenie". It isn't really explained what a "greenie" is.
Unlike Star Rangers, I didn't really recognize this one just from your description, but Wikipedia informs us that Andre Norton wrote another novel which is considered part of the same "Central Control" series. The second one, which I vaguely remember as a science fiction mercenary novel which I read long ago, was Star Guard. (And a little more research finally told me that Star Soldiers is the name of an omnibus collection of Star Rangers and Star Guard, rather than being a third book in the same series, as I assumed at first.)
Wikipedia describes part of the premise of Star Guard as follows:
Information given in the story indicates that Humanity only developed
space travel far enough to attract the attention of Central Control in
the 37th century AD. Norton explains the implied retardation of human
development through references to nuclear wars, which presumably
caused so much destruction that civilization took an extra sixteen or
seventeen centuries to achieve a level of development suitable for
resuming Humanity’s reach for the stars.
Presented in the guise of a history lecture at an alien university,
Norton's introduction explains that in the 40th century the people of
Terra (the Latin name having replaced the Anglo-Saxon Earth) can only
go to the stars as mercenaries. On alien worlds Terrans fight
brushfire wars and thereby help Central Control maintain peace within
its vast interstellar empire.
I didn't remember that much detail about this one (such as which century it was set in, or the name "Central Control," or the reason humans were only allowed to travel to other systems as mercenaries), but I now strongly suspect that Star Rangers and Star Guard are, in fact, the two loosely-linked books which you remember reading so long ago.
