My first thought was that the legislative session for each one of these states ended right before July 1, however, a look at the National Conference of State Legislature's website showed that the legislature in each state adjourns at a different time, although these times could have been different in the late 60s.
What appears most likely is that during the 1960s there was a renewed interest around the country in revising state constitutions. Michigan adopted a new constitution in 1964. Connecticut adopted a new constitution in 1965, and Florida adopted a new constitution in 1968. There were also aborted efforts in the 60s. Maryland, New York, and Pennsylvania all had constitutional conventions in 1967 that failed to be ratified by the voters. All told, 13 states adopted constitutions from 1940-1980.
This activity is no doubt due to the changes undergoing US society during the 1950s and 1960s. Many states constitutions had been in place since right after the Civil War, and thus were not exactly in line with modern day values.
As to whether the three states in question, North Carolina, Illinois, and Virginia, engaged in any sort of coordination the answer is likely, no. The North Carolina Constitution was concerned with reworking the functioning of state government. The Illinois Constitution was concerned with issues of substantive law (abolishing the death penalty), as well as the general inflexibility of the Illinois Constitution. The Virginia Constitution dealt with civil rights issues as well as state bonds.
All three states did have Democratic governors at the time, but the democrats in North Carolina and Virginia were not the same democrats as those in Illinois. Additionally, each state used a governmental committee to formulate what amendments to put forward with the input from interested parties for example bar associations, academics, and concerned citizens. There is nothing to suggest a supra-state type of coordination, especially when some of the changes to the constitutions were extremely state specific.