Different conditionalists have different views on the nature of the soul.  Therefore, it makes it difficult to write and dialogue with them because we would have to find out which particular conditionalist holds to which particular view of the soul.  Nevertheless, the following is a quote that summarizes the three main views.

“Evangelical conditionalists also differ in terms of what we believe the Bible says about the constitution of human beings, and also about whether people are conscious in the intermediate state between death and resurrection. Some are anthropological physicalists or materialists who believe human beings are physical creatures, the functioning of whose minds is dependent upon their living bodies. Others are substance dualists who believe human beings have immaterial souls, but that they lack consciousness between death and resurrection. Still, others embrace a traditional body/soul dualism and contend that the immaterial souls of human beings live on consciously after death (although not immortal in any sense), until a resurrection of the body. The same diversity of perspectives exists within evangelicalism more broadly, and therefore is not a logical requirement or consequence of CI.” (rethinkinghell.com/Rethinking-Hell_Statement-on-Evangelical-Conditionalism.pdf, #8.4)

To summarize the above paragraph, consider the three views of the soul

1. Physicalists/materialists – the mind of the human being is completely dependent upon the living body.  This means that when the human body dies, the human mind ceases to exist.
2. Substance duelists – the human soul is of a different substance than the physical body and can continue on after death, but they lack consciousness.
3. Body/soul dualism – the human soul is of a different substance than the physical body and continues on after death and conscious existence.

Depending upon which view an annihilationist holds to, there will be ramifications with each position.

In physicalism, if the soul ceases to exist upon physical death, then this would require a form of soul sleep (which itself is different definitions) and or complete extinguishing of the existence of the soul. But, this position has problems.  See Problems with anthropological physicalism/materialism which includes

In the form of substance dualism listed above, the issue of soul sleep once again is problematic. Conditionalists typically state that soul sleep is a state of continued existence, but the person is unconscious. The problem is this is not well defined. If it’s not specifically declared and what soul sleep is, it cannot be cross-examined. But, if it cannot be cross-examined, it cannot be defended either. The conditionalists can’t have it both ways. They can’t say that soul sleep is continued unconscious existence without defining whether or not there is in the activity within the soul in this supposed state. If the person says there is no activity, then what is the difference between no activity of the soul and death where that death is nonexistence? If the person wants to state that there is activity, then what is the activity? Is it thought, self-awareness, contemplation, etc.? Again, these are problems for the soul sleep position, and I have not seen this position adequately addressed by conditionalists.  See also, In conditionalism, there is a logical problem with the soul ceasing to exist and then being resurrected

In body/soul dualism where the soul continues conscious existence after death, there is a conflict with anthropological physicalism which says the soul does not continue after death. Furthermore, this is a more biblical view where the spirit continues conscious existence apart from the physical body (Luke 16:19-31; 2 Cor. 12:2). Nevertheless, some annihilationists maintain that the person who continues to exist after death is later judged and then condemned to eternal nonexistence. But, this position has its problems as well. See Philosophical challenges to the conditionalist view of eternal punishment being nonexistence.