You can transfer your data if you still have both your old switch and new system, you have to a game/saves transfer its part of the setup process on the new switch where it'll ask if you want to move data from the old switch.
If you've already sold your old switch then you may be SOL, saves are stored on internal storage (not the SD card, which gets reformatted anyways when moving to a new system) and Animal Crossing doesn't support cloud saves because they don't want people doing weird shit then "restoring" via cloud save, not really sure how thats a huge exploit but apparently big N thinks it is.
do you still have the lite? you are fine if you do, i think you are fucked if its gone. There is apparently a one time exception Nintendo will make and restore a cloud save for AC (which i guess it does in a "hidden" way) but i don't know if they have actually put out a system for that yet
and yeah N is fucking awful when it comes to that kind of stuff, its a little better than 3 years ago but there are all sorts of issues with saves, islands, second players, etc