Yes, we have one overall federal constitution, and each state has its own constitution. The federal takes precedence over the states’.
It’s tighter than the EU, as we have only one national defense, for example. Although each state has its own ‘national guard,’ which are essentially mini-armies that provide for civil defense and disaster relief in their states. However, they may be nationalized into the U.S. Army in certain situations. For example, the Pennsylvania National Guard is routinely sent to conflicts around the world–Kosovo, Bosnia, Iraq, etc. States have mini-air forces, too. For example, Pennsylvania has an Air National Guard. From what I’ve read, air national guard combat pilots are better than U.S. combat pilots because they tend to be older, retired from active duty, and therefore more experience and training.
I assume EU nations can choose to leave the union. That was tried once here–didn’t work out so well.
The bulk of government funding for social services, health services, human services, etc. is from the federal government and is distributed to the states, which then either provide the services directly or pass the funds to county governments, which in turn either provide the services directly or pass the funds to contractors. The rules and regulations that govern the uses of the federal funding are probably thousands and thousands of pages thick. For example, the “manual” that guides the work of social workers in Pennsylvania had been over 900 pages long at one point.
There are some things the federal government can’t do. For example, it can’t set a nationwide minimum age for alcohol consumption. However, the feds can tie an age requirement to highway funding, for example. Since states need the federal transportation money, they comply with the age requirement.
Theoretically, the feds could set regulations for Medicaid (that’s the national health insurer for poor people (mostly)) that specify the money can only be used if states outlaw abortion. Medicaid provides billions and billions of dollars of health insurance. States would find it hard not to comply.