I have a 3DS for me, and one that the kids share. I understand that I can't use purchases from my system on the kids' one. Can I at least sign in with the same account? I don't want to have to make a new account for the kids if I can help it.
Unfortunately, you can't.
As Nintendo's help page about the Nintendo Network ID states:
You can only register a single Nintendo Network ID to each Nintendo 3DS system. It is not possible to register multiple Nintendo Network IDs on a single system.
A Nintendo Network ID registered on a Nintendo 3DS system cannot be registered to another Nintendo 3DS system. If you wish to transfer your Nintendo Network ID and balance to another Nintendo 3DS system, this is only possible by performing a system transfer.