Sounds like an empty clock battery. Many devices, cameras included, have a little battery (not the main, rechargeable one) that maintains the settings when the main battery is changed or empty. The little battery powers the memory in which these settings are stored. When it's empty, you may find that the camera 'forgets' settings like the date or the last used manual exposure settings, ISO etc. as soon as you take out the main battery.
There's a thread about the issue, here:
https://www.dpreview.com/forums/thread/4642436
Probably some more if you search around a bit.
There are a couple of ways these auxiliary batteries may work. Back in the old days, they were (at least in other equipment than cameras, such as computer mainboards) often rechargeable NiCd batteries. These would often start to leak after a time (spilling gunk including some cadmium onto the electronics!) and had to be replaced. In the 1990s/2000s, they were often non-rechargeable lithium button cells, which had a service life of several years, and sometimes in excess of a decade - but they ultimately fail, too. It's also possible to use so-called super-capacitors in this function. Theoretically, these would have an infinite lifetime, as they act (sort of) as a rechargeable battery.
In most equipment, these auxiliary batteries are not user-replaceable. They require partial disassembly of the product and sometimes soldering to replace it. A suitable spare part also needs to be sources, which may involve some rooting around to locate the proper part.
The TL;DR is of course that sending it to Nikon for servicing is a logical course of action. It may not be the cheapest solution, but it's certainly a low-risk one.
