Your question looks straightforward but it is not so easy, indeed !
Ken Rockwell’s overview is quite exhaustive. Basically, AF-S and AF-I would be a good match for both your FE10 and your D5600. Most if not all AF-I are telezoom/telephoto. Furthermore, no AF-S with E, no AF-S with G of course would match your needs as they have no aperture ring. For instance, AF-S (“Silent motor”) can be with or without ring aperture. In the latest, G or E are added. Conversely, most AF-D have no internal motor. Autofocus can’t work with your D5600 without an internal motor. But some D lenses have silent motor inside! They are “D type AF-S”. That’s the kind of lens you will need. Yes, it is somewhat complicated, I agree.
This is confirmed by Nikon official pages about FE10 (what works for FE10 will work for FM10 AFAIK) and D5600 lens compatibility (open on the D5600 page the xls file). It is confusing to state FE10 works with AF-S. It works with AF-S non-G and non-E !
The FE10 will require a full frame lens, so no DX lens for your requirements, only FX lens would be a match.
To summarize, you need an FX lens which is D type with AF-S.
When it comes to the lenses you want, the choice looks very limited and a little bit… awkward. Indeed, most of the AF-S and AF-I which are no G or E are Telephoto / Pro super Telephoto. So, I went through the whole list of Nikon lens to double check (again look at the xls file in reference above) and the 2 only lenses I could find were:
AF-S Zoom-Nikkor 28-70mm f/2.8D IF-ED
AF-S Zoom-Nikkor 17-35mm f/2.8D IF-ED
Both zooms used to be excellent at their time (pro zooms) but they are outpaced, whilst very solid and very heavy. You can still buy the 17-35 according to Nikon official site. And it is easy very easy to find the AF-S 28-70 D second hand, but usually quite expensive too. Again, both zooms are pro lenses.
To be honest, I understand the need to have the same lens for both bodies, but I am not totally convinced it is the best approach due to the limited and expensive choice you have. Well, at least you know what is possible.