As far as the courts are concerned, you are a "resident" of only one state at a time. Residency, for jurisdictional purposes, means physical presence with the intent to remain. Thus, if you have lived in NY your entire life, and travel to Georgia to visit friends and family, you remain a resident of NY despite your physical presence in Georgia. You can stay in Georgia for quite a long time, in fact, so long as you intend your stay to be temporary, i.e., you intend and expect to return to NY at some point in the foreseeable future.
If you intend to remain in Georgia, however, then you become a "resident" of Georgia for jurisdictional purposes as soon as you cross the border into Georgia.
What this means is that you are subject to the laws of New York, and reachable by its courts, wherever you may be for as long as you remain a legal resident of New York. You are also subject to the laws of Georgia by virtue of your physical presence there, even though you may not be a legal "resident" for jurisdictional purposes.
If you have a right to something under the U.S. Constitution, however, a state cannot deny that right to you on the basis that you haven't been a resident for a set period of time. For example, laws that deny you the right to vote until you've been a resident for x number of months have been declared unconstitutional. The rationale is pretty simple. If you were a resident of New York, and move to Georgia on Tuesday, June 7, with the intent to stay, then as of June 7 you are no longer a resident of New York and are now a resident of Georgia. You can no longer vote in NY because you are no longer a resident of NY. If Georgia denied you the right to vote until you had lived there for six months, then you would not be able to vote anywhere during those six months. Hence, such laws are void. Similarly, States cannot deny things like driver's licenses to new residents.
That's the theory, anyway. In practice, the states impose difference restrictions and requirements for different things, some of which you mentioned. It has been held, for example, that States can impose minimum residency periods before granting in-state tuition benefits.
There is no single answer to your question. It all depends on what you want to do, what your intentions are, and whether there is sufficient evidence to support your claimed intention. I would caution you against trying to maintain "dual residency", however, for a number of reasons. First, if you are successful you subject yourselves to the laws and jurisdiction of two states. That means you are also liable to pay taxes (including income taxes) in both states. It's o.k. to switch your residency back and forth periodically (even once or twice a year), but you should officially be a resident of one or the other at any given time, not both.
Second, you may not pull it off. If you think you are a resident of say, Georgia, and claim rights under Georgia laws that protect Georgia residents, the courts may ultimately disagree and find that you're really a resident of NY who is temporarily staying in Georgia. Poof -- your Georgia protections have vanished and you are probably in violation of those laws.
