It's handy that the agency's initials didn't change, but that the is a problem. The Veterans Administration? Yes. That's how people talk. The Veterans Affairs? No.
To review, the general guideline is that we use the with initialisms when both of the following are true:
In "Lapsing Into a Comma," I point out that this means the Bureau of Alcohol, Tobacco and Firearms gets a the with its common BATF abbreviation but not with its more common ATF abbreviation. It's "the Bureau ..." but not "the Alcohol ..." (The agency has since become the Bureau of Alcohol, Tobacco, Firearms and Explosives, but it's still called ATF.)
Now we have a similar case with VA and, uh, VA. What to do? One answer is to use the abbreviation only adjectivally. As a second-reference noun, use Veterans Affairs or the department. But that can get unwieldy in a story with several such references. Or you could call it VA without the the. The irony, if I may risk a rare use of that term, is that people drop the the in agency references that need the word. Things happen "at EPA" or "at OPM." Sic. Sic. But the tradition of "the VA" dies hard.
A corollary problem is the first reference to a VA hospital. That second reference works fine, but, if you want to get picky (and of course we do), what used to be a Veterans Administration hospital is a Veterans Affairs hospital only if Department of Veterans Affairs was used earlier. Otherwise, you really should call it a Department of Veterans Affairs hospital.
Veterans hospital, anyone?