There are some obvious 'quirky' things an adjective could be made to do, e.g. itself be in a strange case or cause the noun to be in an unusual case. However, we can also imagine other things.
Certain adjectives could force their noun to be in some case, or at the very least block one case from being marked. Maybe something like
nom → acc
acc → acc
dat → dat
abl → abl
Another option would simply be that the adjective never occurs in NPs of a certain case. This has two possible interpretations: 1) such NPs mark a different case instead, or 2) such NPs simply are never used in positions/constructions that take the forbidden case, and some synonymous phrasing is used instead.
Further, we can imagine quirks in certain case marking positions. Consider, for instance, object complement adjectives, e.g.
I painted the house red
One could imagine that certain adjectives require special marking that other adjectives do not. And here, we could have a slight bit of alignment shenanigans appear - maybe some adjectives, such as 'dead' follow an ergative alignment,meaning that they take the nominative|absolutive when applied to objects or to intransitive subjects. It's not common in English for transitive subjects to take adjective complements, but maybe your language does.
The example clause of painting a house red suggests to me a different thing; we'll stick to painting for now, I hope the reader is able to re-apply the idea to other topics. Maybe basic and non-basic colours (for some way of dividing colour-space up) take different markings:
I painted the house redI painted the house of orange
The question with regards to the language then is whether this is specific to the combination of the verb "paint" and a set of adjectives, or whether it's just specific to adjectives in that position in general, e.g. would something along this line also have 'of' or not:
I found it (to be) orange
Both ways are reasonable in a language with this kind of marking, and one can probably imagine different subtypes of complements that a conlang could have acting differently, classified by aspectual or volitional or kinetic features or whatever.
2. A Vaguely Alignment-like Thing for a Marker
In e.g. Sami languages, adjectives have a thing that isn't quite congruence, but is not far away from it either. As attributes, they take a suffix, as complements they do not. Thus "the red house" has the marker, "the house is red" does not. We could now imagine situations where this is broken, or even having some adjectives go the other way around, maybe even introducing some kind of 'alignment-like' way it works.
Consider, for instance, adjectives that take this marker whenever they have any kind of NP as complements. I'll use -X as a shorthand for the morpheme in the examples:
the big-X man
"the big man", because big is an attribute.
the man is big
"the man is big", because big is not an attribute
Now, 'afraid' is not really used as an attribute much in English afaict, but let's pretend:
the man is afraid - not an attribute, no nominal complement
the man is afraid-X of spiders - not an attr, but does have nom. compl.
the afraid-X man - a compl.
the afraid-X of spiders man - attr., as well as nom. compl.
to be afraid-X of spiders is a common phobia - nom.compl, but not an attr.
to be afraid is counterproductive - not nom.compl, not attr.
3. A Dummy Head for Adjectives
Let us imagine a situation where adjectives take case marking when attributes, but never when complements. Now, this language has, historically, developed a need for case marking on attributes on occasion, but the restriction still exists. A dummy NP head has turned up that does carry case, though. However, this dummy head is defective, and lacks the nominative. Now, we can imagine situations where adjectives still are needed as subjects, and we can further imagine that, say, the accusative form of the dummy head turns into a nominative-accusative case form. However, we can also imagine a situation where voice operations are used to turn the adjective into a permissible subject without introducing an additional case form to the dummy head.