UPDATE: My problems may have been my fault.
I happened to notice that I did not change the power adapter when I switched in the 3782 router, so I replaced the adapter with the new one and my problem was resolved.
It is beyond my ken to understand how the normal internet and VPN connections worked, but some remote drive mapping assignments on the VPN were KO'ed
The old adapter output was 12v @ 1A and the new one was 12v @ 1.5A.
Hope no one else has to learn this lesson the hard way !
So, by and large, I'm happy with the 3782 for now, except for the DNS issue.