If the DVRs are compatible, it is possible for you to move them to an external hard disk (EHD) and move them back yourself, but I've never heard of Dish doing it for you. (It will take hours - not something they or probably you would want a technician hanging around for.) If you still have the old DVR, it may still be possible to do this. If your DVR was truly obsolete, then it may not be compatible.
Also, keep in mind that on an old obsolete DVR, that disk could have crashed any day now anyway. Shows sitting on a perpetually spinning old hard disk are a convenience, but you should never think of that as archival storage.