Your best move is to contact a Georgia attorney ASAP. For a listing of Georgia Elder Care attorneys, you can take a look at http://www.eldercarematters.com/georgia/.
Generally, documents that are valid in the State where they are signed will be deemed valid in other States, even if the other States impose different or additional signing requirements (such as whether they must be notarized or whether they need to be witnessed, and if so, by how many witnesses). Thus, the Missouri documents may be just fine for Georgia purposes, but only a Georgia-licensed attorney can answer that question with authority.
The fact that your father has Alzheimer's may or may not impact his ability to sign documents. Obviously, mild Alzheimer's would not be an insurmountable problem, and even more advanced stages of the disease may not preclude a person's ability to sign documents that set out his or her simple choice of who will be the person's surrogate decision-maker. What level of capacity is required to be able to sign different kinds of documents is a question of State law, so again, you are best advised to talk with a Georgia attorney at first opportunity.
If your father's Missouri documents turn out to be invalid or unhelpful in Georgia, there may be a variety of things that can be done under Georgia law to protect his choices as previously set out in his Missouri documents. For example, if your Georgia counsel reaches the conclusion that the Missouri documents don't work in Georgia, you may be able to go before a Georgia court to have a guardian appointed for your father. Your father's expression of his choice as to who would make decisions on his behalf if he became unable to make them himself would, at the very least, be grounds for the Georgia court to appoint the very people your father had hand-picked while he was competent.
You can find Hawaii-specific information about these questions at www.est8planning.com.