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I was really anticipating the Z6 mirrorless camera, but the limited selection of Z Mount lenses is kind of disappointing. And it doesn't make sense to me to start investing into F Mount lenses if I end up with a Z6.
The question is, which body do I want? The D7200/7500 would allow my girlfriend and I to share lenses. As would the D500 if I thought I needed the extra horsepower. I think that I'd be happy with the 16-80mm for the vast majority of the pictures I like to take, with maybe the 200-500mm for wildlife.
The Z6 with the 24-70mm would leave me wanting more reach, and 3 lens systems to juggle. The A7iii with the 24-105 is pretty tempting too. My girlfriend will be happy shooting DX due to size and weight and I'm by no means an expert and still have lots of learning through experience to do.
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In short, for maximum in simplicity between His & Her camera kits, stick with a crop sensor. The D7200 has good pro-level features that I think would allow you to expand your skills faster than the D5xxx series (based on my own experience). However, if you have the budget, I think the D500 will give you more service life.
Camera selection, in my opinion, really boils down to two major factors: budget and type of photography (or intended use). Of these, I think your intended use is more important when deciding between sensor sizes (because you can spend a range of budget on each sensor class) and I think Zack is on the right track with, in essence, zeroing in on the type of photography you expect to do. If you're like me (a generalist, but mostly landscapes), full frame provides greater flexibility. If you prefer wildlife, it's hard to beat a quality crop sensor camera. Based on your comment about the 16-80mm, a full frame with the 24-120mm would work well (comparable field of view to the 16-80mm on a crop sensor) - and the 24-120 would function just fine on the D5600 (with a different field of view when mounted on the crop body).
Ogive provided a bit of a GD-style response, but he's right in one regard: Mirrorless is the future. However, the future is not quite now (with apologies to Nam June Paik). It will be several more years before mirrorless closes the various performance gaps with DSLRs and if you're the least bit annoyed with having multiple camera systems to track, adding Sony to your current Nikon mix will complicate life more than help. That said, I've played with a Z7 (at a photo conference, not a photo trip) and it is impressive. If you're seriously considering the Z6 but are concerned about the lens options, the D750 is still a very viable choice with a bit lower price.
Re: Z-mount lens selection - Remember that Sony's native lens selection for their FF mirrorless bodies was very slim when they introduced the lineup and Nikon have published their roadmap showing a plan to have 12 Z-mount lenses within 2 to 2.5 years. I think Nikon will have at least one more mirrorless body released in that timeframe, too. My point is that adding a Z-mount body to one's kit today is definitely setting up for the future and most F-mount lenses are fully functional via the FTZ adapter. If I had not acquired my D850 just a year ago, I would most definitely be looking at that Z7. It's a sweet bit of kit.