Swapping disks from ReadyNAS to ReadyNAS NV+

As long as it is an NV+ V1 (Silver in color, looks like your NV but with the addition of the LCD panel), then all is well.  If there is a sticker on the bottom that says V2 or even V3, don't let that bother you.  A "real" NV+ V2 is black in color, has an external power supply, and says V2 on the front; and that won't work.  Since you mentioned swapping the power supplies, I think your replacement is the right type.

Since you gain the LCD panel and the unit is going to be at least a little newer, I'd put the drives from the old into the new unless there are signs the old one was not well cared for (like excessive dust collection).

Assuming you have kept your firmware version up to date, you should first put any old scratch drive that contains no data you care about into the unit and power on, let it build a volume (which should be quick with one drive), then insure the firmware is up to date or update if needed.  Then power off, remove the scratch drive, insert the drives from the old unit (in the same order is best), and power on.  As long as the volume wasn't corrupted by the power supply failure, which is unlikely if the failure occurred at power-on as it usually does, then you should be back in business.

For extra assurance the replacement unit is fully functional (it is, after all, still pretty old), you could power down, swap the scratch drive to the next bay, and power back up.  Do that going through all bays before you put in the volume from the old one.