Hey there -- if the story "event" mentioned above where you are forced into choosing between catching a pet or ignoring it hasn't happened yet, and Jinxie is still telling you (in the bottom right corner) that you need to try catching a pet, you need to keep exploring until it happens.

Also, in order to add a pet to your party, you need to pay to expand the number of available slots. You can do this in Esior's Haven.

I hope that helps!