I finally got Baabara to the part of moving, but she doesn't ping me about it. I restarted the game, swam out of sight, of the other villagers for 5 minutes and still no ping!
Do you mean you heard from another villager that Baabara is moving? If you want her to move, your best choice would be to not talk to her at all anymore; she will still move out in five days even if she hasn't pinged you about it.