To answer your first question, the Constitution would become a document that people no longer use. Just like the Articles of Confederation or any other outdated document that is no longer needed. The purpose of the Constitution is to enumerate the powers of the federal government, so without a federal government, the Constitution has no purpose or meaning.
As for police, you'd do the same to protect yourself from them as you would for anyone else trying to wrongfully imprison you. Most police, though, derive their power from municipality, county, or state governments, not the federal government. If the federal government was dissolved, then any federal law enforcement agencies, such as the FBI, ATF, etc., would cease to exist as well, so, if any individuals formerly tied to those organizations tried to attack you, the question is: What would you do about it?
Your questions about healthcare all seem very contingent upon the assumption that money will continue to operate in the same way. Let me assure you that without the constant threat of military force, the dollar will completely collapse. As it should. Its value is derived from nothing but the threat of violence. The collapse of the dollar is inevitable, though, in any case. We are a nation of debtors, printing money with no backing to pay those debts. If you understand economics, you'll see that this is not a trend that can continue indefinitely. Sooner or later, any sort of savings you have that are in dollars, or dollar-backed currency, will lose their value.
How will that affect healthcare? Well, it depends on the general quality of morality that your local heathcare providers have. I suspect that one side effect of this will be that a 20-minute ride in an ambulance will cost significantly less than a year's earnings, as will many other currently expensive medical procedures. Most of the current problems with our healthcare system are tied directly to the insurance companies (that's why the ACA, which requires everyone to be their customers, is so absurd). So, currently, every time you pay for healthcare, you're not just paying for the services you've used, but also for many layers of bureaucracy within the healthcare system and the insurance companies. We have leeches sucking profit out of the system without actually contributing anything to it; without them, everything will be significantly less expensive.
The federal government and, by extension, the president, have no power to dissolve state governments. What states do after that is entirely up to the people of each state to decide for themselves. The purpose of Adam's plan is to move the bases of power closer to the people, starting at the top. The few legitimate purposes that the federal government serves are, largely, redundant, which is why it can pretty safely and quickly be dissolved.
I think that it will take much longer to move away from state and local governments in a way that will be minimally harmful to people that have become reliant upon government for their basic needs. Before we can safely do away with the governments closer to us, we need to ensure that our neighbors are ready to take care of themselves, which will take time.
Does that all make sense? Do you have any other questions?