It reforms in the Abyss
Nothing states that a shadow demon itself cannot reform, only that it is created by the failure of another demon to reform.
When a demon's body is destroyed but the fiend is prevented from reforming in the Abyss, its essence sometimes takes on a vague physical form.
(Monster Manual 54)
However it is still a demon and when a demon dies...
...the fiend dissolves into foul ichor. It then instantly reforms in the Abyss, its mind and essence intact.
(Monster Manual 52)
A shadow demon remains both a fiend and a demon so this rule applies.
If the demon is still prevented from reforming (such as by the same effect as the original demon), the essence of the shadow demon could become a new shadow demon or not since a demon that can reform only "sometimes" assumes the form of a shadow demon.
The Monster Manual answers this, explaining what you turn into based on where your soul goes. Being killed by a succubus doesn't change your destination, it's the fact that you have an evil soul that drags you down.
When the soul of an evil mortal sinks into the Nine Hells, it takes on
the physical form of a wretched lemure. (p67)
If you wish to become a low level devil or other being, you may need a promotion.
Archdevils and greater devils have the power to promote lemures to
lesser devils. Archdevils can promote lesser devils to greater devils,
and Asmodeus alone can promote a greater devil to archdevil status.
This diabolic promotion invokes a brief, painful transformation, with
the devil's memories passing intact from one form to the next.
So, simply be useful, and you may be promoted up from being a gooey, clay mess.
Alternatively, if you go to the abyss, you may be turned into a Mane.
Spawn of Chaos. The Abyss creates demons as extensions of itself,
spontaneously forming fiends out of filth and carnage. Some are unique
monstrosities, while others represent uniform strains virtually
identical to each other. Other demons (such as manes) are created from
mortal souls shunned or cursed by the gods, or which are otherwise
trapped in the Abyss. (p50)
Your form here is elevated by how much blood you spill, and whether others promote you.
If instead you go to Hades, you become a larva.
The layers of Hades are called the Three Glooms — places without joy,
hope, or passion. A gray land with an ashen sky, Hades is the
destination of many souls that are unclaimed by the gods of the Upper
Planes or the fiendish rulers of the Lower Planes. These souls become
larvae and spend eternity in this place that lacks a sun, a moon,
stars, or seasons. Leaching away color and emotion, this gloom is more
than most visitors can stand.(p63, DMG)
While in Acheron, you fight endlessly.
Acheron has four layers, each made of enormous iron cubes floating in
an airy void. Sometimes the cubes collide. Echoes of past collisions
linger throughout the plane, mingling with the sounds of armies
colliding. That's the nature of Acheron: strife and war, as the
spirits of fallen soldiers join in endless battle against orcs devoted
to Gruumsh, goblinoids loyal to Maglubiyet, and legions assembled by
other warmongering gods. (p66 DMG)
Best Answer
If the first contract has the price of the soul, the second contract cannot be entered into because the character no longer owns the soul and thus cannot sign it away.
Descent into Avernus (a published adventure) has the necessary rules for infernal contracts.
The contracts are described as:
That last paragraph in particular does mean that effects that would attempt to "capture" the soul at the moment of death (like soul cage) are unable to divert the soul, as it is bound to hell, and this is enforced by the multiverse itself.
The basic form of a deal, and how it's contracted is described under "Making Deals":
The restriction on a soul being claimed twice is described under "What Devils Want > Souls":
A characters soul being forfeit is a common penalty clause of an infernal contract, and when the character dies they are instantly reborn in the Nine Hells as a lemure (no chance of revivify etc). This is described under "Infernal Contracts > Breach of Contract":
It is however, possible to have a contract voided with the agreement of both parties:
A devilish contract can also be destroyed, and voided as a result:
The text goes on to describe different contract types and how they can be destroyed.
If neither contract has the soul as the price, but merely as a penalty clause, then it doesn't matter which contract they violate, the Nine Hells still gets its due.
The soul is pledged to the Nine Hells upon violation, not the individual devils. As a result, it doesn't functionally matter which contract they violate, they still forfeit their soul, when they die their soul is still transferred to the Nine Hells, and they are still reborn as a lemure.
This is described under "Infernal Contracts > Breach of Contract":