Well, it appears the DNS nameservers did not affect or fix the issue it was intended. It turned out to be a conflict of plugins (most likely some javascript conflict between the two plugins as discovered here) between flv-embed and lightview-plus.
I don't see any harm in keeping all 6 nameservers, so they will all stay for the moment.
I've taken the original nameservers provided by the webhost off our records again.
I used intoDNS and found that the SOA serial being provided by everydns and 3ix were different. There was mention of a possible conflict that may produce unwanted effects. Also, the SOA record retrieved (when it chooses the 3ix nameserver to be its primary one) showed an expiry time that may be too long – that triggered the warning.
So now, I'm just using everydns and the records look okay now.