Glad to hear things are still working properly with the D600.
As for the D610, I'd suggest we open a new thread under the Hardware, Components and Peripherals
section of the forum.
With what you are describing, it sounds like it may be a hardware issue with your DVD drive.
Let's start a new topic there and see what can be determined. It sounds like an issue with your DVD drive itself but someone may have some ideas/suggestions. Either way, if what you said is true (that the DVDs burn, but just do not read correctly on that same system) then you at least can still burn DVDs/make all your necessary backups, but will just need to verify/check them on a different machine.