Of course, I'm not from Virginia....so I can't answer this question as a Virginian would! But I think it is a very good question....and I don't have a "pat answer"...only some thoughts regarding the question. And even though the question is NOT about the War between the States.....I'd like to try to answer within that context. (Because I think that was a defining period in American history.)
Under the Constitution, the States had all rights reserved to them.....NOT specifically granted to the Federal government as outlined in the Constitution. Prior to the Civil War it was common to refer to "these United States". After the civil war....it became common to refer to "the United States". So I do believe there was a common perception change as a result of the civil war.
If I had been Robert E. Lee.....when forces of South Carolina fired on Fort Sumtner.....I believe I would have also declared my allegiance to Virginia...and what would eventually be the Confederacy. He was raised in a time that State's rights were still believed to override federal authority....and that question had been raised many times preceeding the actual secession from the Union by South Carolina. (It had actually been threatened...enacted and then withdrawn.... by South Carolina under the Presidency of Andrew Jackson...a Southerner himself. He flat out declared that if S.C. sececceded...Jackson himself would march an Army to the South Carolina legislative body...and execute the treasonist.) But in short....the question of the legality of secession was discussed, professed, written about, and frequently threatened....but never "put to the test" prior to Lincoln's election. In the minds of Lee and men like him.....he was a Virginian. And as a Virginian he was part of the United States......but he didn't think about himself as "an American".....like we think of it today!
If I had been Abraham Lincoln......during the sececcion....I would have held that the Union, and the institution of Union, was preminent over states rights. And my line of reason would be that in fact upon ratification of an "entity greater than the individual states themselves.....the union of said states" the union became more important than it's individual component parts. And if a Republic was to work.....it had to exclude the right to withdraw from that union each and/or every time the specific component parts....did NOT get what they wanted. In other words...compromise would always be necessary at some degree and level.....or we would eventually be no more than a collection of individuals....working together only when it personally suited us! And that by acceptance of Union during the ratification.....the States had become a part of something bigger. And just as an "arm did not have the right" to leave the body....of it's own accord.....neither did individual states have the right to dissolve "their union".
I also am not from the South....so I can't honestly answer that portion of the question either. Though I've known many fine people from the South....and without a doubt....the majority were proud of their "Southern heritage"....but considered themselves to be "good Americans first". (Maybe because the majority of southerner's I knew....were in the military....and took pride in being American?!? Of course a good way to start a "good argument" would be either a Yankee or a Rebel....to comment on one another's heritage

!)
Because I can't answer the question as asked.....I'm neither a Virginian or a Southerner....I'll answer the question for myself.....within the context of the question's intent. (Or my perception of that intent.)
If I know someone....and they tell me they are......African-American, Hispanic-American, Korean-American, Irish-American, etc.......I usually try to ask inoffensively.......
Which part of the answer do you think is most important to you? Or most descriptive of you? In other words....is the fact that you are an American the most important part of that to you.....or the least important part to you? Frequently I will get an answer that is sometimes ambiguous...and sometimes I will get a direct and open answer. And sometimes they don't believe the American part of that answer.....is the most important part. (And that always disappoints me a little.) Because I truly believe it is because we were fortunate enough to be an American......that allows so much latitude to recognize any or all of the others. And I don't believe we should give up our traditions or cultures....but I do think it is important that we recognize why being an American is so much more than just the sum total of its parts!
Being from a northern state by birth......and where I grew up.....I will say this also. When I see a "rebel license plate" or a "rebel decal" in the back of a pick-up truck window......on a Pennsylvanian's auto......I think it says something different to me then when I see that same emblem when I'm in the South! And case you are wondering why......it is because a Northerner can't claim it as heritage. Therefore it says something else to me about motivation. Now it may just mean they like pick-up trucks and Country music.....but in all honesty.....it frequently means something else. And I base that on my past experience.
A Southerner can claim their heritage......and I don't assume anything negative about that person. Unless they prove to me otherwise. But when a "Yankee" is displaying that.....it frequently is not a "positive statement". (And if you are a Yankee displaying that....and what I just said doesn't include you.....I'm then NOT talking about you. I'm only addressing those that it does apply to!)
Just my opinion!
P.S. And I know this doesn't exactly count....but......
1. I'm an American.
2. I'm a Pennsylvanian, (at least until I get the heck out of here and back to Alaska).
3. I'm a Northerner.
In that order!