TL;DR: No.
What is a soul?
Some have observed that we can't answer this question without first addressing the question "what is a soul?". While there is some truth to that, I don't believe that answer is so ambiguous or lacking in consensus as to cause a real issue. Therefore, I will define "soul" thusly; that part of a person, spiritual (that is, immaterial) in nature, in which volition resides. Often the soul is believed to incorporate some aspect of memory and/or personality as well, although this seems debatable even by those who presume that souls exist. Additionally, it is unknown what abilities (if any) a disembodied soul
possesses.
To those wondering about the nature of souls, I would recommend Orson Scott Card's Enderverse series, as his notion of "philotes" offers a plausible-seeming explanation as delivered from a more scientific, rather than theological, perspective.
What is the argument?
It should be noted that there are a variety of similar "attacks" on the soul of this form, with "damage-induced personality change" being a leading factor. Harry Potter and the Methods of Rationality also explores the question (from a Materialist perspective) and comes to a conclusion which I disagree with. In general, the question isn't limited to just Alzheimer's.
What shall we conclude?
For the sake of argument, we'll assume that the soul exists. The inability to explain the semblance of free will (whose existence is also sharply debated) in purely material terms would seem to argue in favor of this existence, but since the Question asks whether the claim of existence is undermined, for our purposes it is satisfactory to take the soul as a presupposition.
The main issue with arguments that "X disproves the soul" is that they tend to assume that the material realm cannot affect the spiritual realm. If this assumption is false, it's easy to see how physical (material) trauma could "spill over" into spiritual trauma, with corresponding effects on memory and/or personality, even if memory is a spiritual phenomena.
It seems more plausible, however, that the soul is only part of an intricately connected system, of which the material brain is an integral part. It seems beyond question that the brain (and even neurons outside the brain) play an integral role in processing both input and output signals. A person with no eyes, for example, can't see, and a person with nerve damage can't move their limbs. This, however, in no way invalidates the existence of the soul; it only indicates that the action of the soul is mediated by material substances.
Moreover, if we take a person's actions to be a combination of the material function of the brain and the soul, we can also see how chemicals or physical damage could alter a person's behavior, even if the soul exists. A good analogy might be to think of the physical person as a horse, and the soul as a rider. The rider "steers" the horse and provides higher-level guidance that what the horse can achieve on its own, but the horse also has the ability to override or ignore the rider. Even from a strictly material sense, we see this as in the case of reflexes wherein the body acts without instruction from the brain.
Consider also the brain as the "house" of the soul. Just as one's external environment influences behavior and mood, so too might brain chemistry "propagate" to influence the behavior of the soul. This seems especially relevant if we assume that certain functions, such as maintaining separation between one's "inner voice" and one's tongue or accessing memories, involve material function of the brain (hence explaining why chemicals can "loosen inhibitions", or why material causes can influence memory retrieval). Imagine, also, the frustration one must feel trapped in a body and mind that are not functioning correctly, and how the resulting frustration and sense of dissonance could influence personality.
Therefore, the long version is that such phenomena must certainly influence how we view the (two-way!) interaction of the soul with the physical world, but they do not preclude the existence of the soul.