Virginina secedes for the same reason as OTL. They perceive secession as a right which they don't initially care to exercise. Once the Union Army is called up, the central government is using force to crush people's legitimate rights (the views of 19th century Virginians, not my own). Time to rebel.
The Constitution would get written, much as OTL; Toombs basically wrote it around 1835 and has kept it in his desk drawer waiting for the right time. The perception that secession was some sort of spontaneous reaction to the events of the day is flatly untrue; after the nullification crisis of 1833, the smarter southern firebrands understood that that was their warning shot - the next time, it would be war. They have been preparing for it for 27 years OTL, 17 years TTL.
I can imagine some prominent failures to ratify it, though. Texas leaps immediately to mind.