"This shirt is not old."
this is pretty old and you probably already know this stuff, but です is not a verb, is a copula. They word differently.
Also in this particular sentence です is being used as a way to soften the tone into one that is politer, so the structure is このシャツは古くない and the です is there just for politeness sake.
Another thing that you probably don't know is that Japanese verbs don't really conjugate, they use auxiliaries to create a pseudo-inflection,
in the case of 古くない you are changing 古い into 古く in order to connect with ない and negate the concept of "old".
in the case of 古くではありません、you indeed can say this but is not natural and not common, it sounds like old Japanese if I'm not mistaken. And you are using 古く as a pseudo-noun, this is not possible with 古い。
However, you can also say 古くありません and that's exactly as 古くない、you are connecting 古く to ～ありません。This means "not old" in a more formal tone.
How come we make the adjective negative instead of the verb?
so after that, it seems logical that we add the ～ない into the connective form of 古い、because です is not a verb nor is part of the grammatical structure in this sentence. Usually happens with other i-adjectives as well.
Or how exactly does the conjugation and the relation between the adj and the verb work?
there is no relation, there are two different types of predicates, one is with i-adjectives as シャツは古い and the other type is with the negative counterpart of です and the noun 古 like シャツは古ではありません "the shirt is not old", this is not a common way to phrase this though.