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I've recently come across a dilemma of whether to capitalize a plural noun, given that it's described by a proper adjective (like a place) and a common adjective (a descriptive adjective).

Example-

...big and Saharan deserts

Or

...big and Saharan Deserts

In both of these cases, we're talking about multiple deserts, and one is big, and the other Saharan. Does desert get capitalized? Would the order of the adjectives matter?

I know it's not a great example (I'm not talking specifically about the Saharan Desert, but any proper adjective would do), but can anyone clarify? I've never learned about this specific topic (with one proper and one descriptive adjective). Also, if a question like this exists, please kindly direct me to it! Thanks!

You'll find both initial caps and initial lower-case:

From Handbook of Landscape Archaeology, B David and J Thomas (eds),

... the linkage between the Arabian Desert in the east and the Sinai
and Saharan Deserts in the west,....

From Great Walls and Linear Barriers by P Spring,

In the semi-arid northern borders of the Arabian and Saharan
deserts

I haven't been able to find a conjunction of Saharan with anything but another adjectival form of a place name, but the preponderance of the Ngram viewer examples seems to be lower-case deserts. The probable reason is the the Sahara Desert is a proper name, but Saharan deserts isn't. Rather it's a descriptive term for the various so-called xeric subregions of the Sahara Desert, which have their own proper names, e.g., the Grand Ergs of Algeria.

This discussion illustrates the fact that the adjectival form of a proper noun keeps its properness (and thus its initial cap) for itself, but also that the properness does not extend to phrases that are proper with the noun form. Thus we learn from wikipedia that

The Mississippi Band of Choctaw Indians is one of three Federally
recognized tribes of Choctaw Indians.

But we would say that

The moundbuilding, Mississippian bands of Indians formed an urban and suburban culture that flourished between 800 an 1600.

In the former case Band is part of the name and so is capitalized. In the latter case, band is not part of the name and so is lower-case, unaffected by Mississippian, which simply acts as an adjective no different from its companion, mound-building.

While I learned that the use of multiple Xs requires a lower case x, (e.g., Mississippi and Missouri rivers), to my surprise I found a page that suggests just the opposite, too:

"...Plurals: When two or more place-names share a common element, the common element is capped: the Thames and Avon Rivers."

To me, that looks wrong, so it's likely a less frequent style.

The AMA Manual of Style says, "If a common noun is capitalized in the singular, it is generally not capitalized in the plural. Atlantic and Pacific oceans Kennedy and Eisenhower expressways Mississippi and Missouri rivers Compass directions are not capitalized unless they are generally accepted terms for regions."