Nintendo Switch Joy-Con (L)/(R) Neon Pink/Neon Green
you can connect these with the switch lite, but you do not need them to play animal crossing new horizons
The straps are included.
My Joycons do the same thing, try updating your controllers in the system settings. If that doesnt work, then you may just need to buy new joycons, thats what I might have to do as well.
you can each play with a single joycon (played sideways) or you can each play with two (one in each hand) so its up to you either two or four joycons. (or you could even play with three one person with two, the other with one)
No unfortunately just joy cons