Choco doesn't ever actually stay at Chocobo 'Paradise' (which is the right name I believe) - and the bit where the fat chocobo says that choco should go on an adventure actually involves a long sidequest where you must visit every beach area on the world map. Once you get to each one get off of choco and you will here a chime, then get back on and repeat the process for every beach. On the last one something will happen! From then on - every time you visit any beach you will be healed fully with all status ailments cured.

Unfortunately I never did this 'adventure' so if you do it something might happen when you visit fat chocobo again!!