Using that logic our secession from Great Britain would be illegal and any Constitution derived from obtaining our independence would be illegal, correct?
Yes - and no.
Let's put it this way. You're a guest at my house (I hate using these, because SOMEONE gets all worked up over precision in metaphors). You ask my permission to leave, and I refuse. You go out the door anyway.
Someone asks "did you have Sam's permission?". Of course you didn't. You left anyway, but you don't go around saying you received it.
Of course it was "illegal" to break away from England. It's not as though countries have gentleman's agreements on how to conduct wars. It's not a country club affair.
Texas vs. White makes it clear enough that states did not enter into something like a marriage with the rest of the United States, free to legally separate at will. Not any more than St Mary's county has the legal right to separate from the rest of Maryland. The counties have separate laws and governments, but it doesn't make them a loose confederation. They're bound, and according to the Constitution, states are as well, forming a perpetual union.