It's called "jellybeaning", and it's a common problem with DPSS lasers. Do a search here on PhotonLexicon and you will find lots of posts on the topic.
You can try to adjust the TEC circuit, as thermal instabilities often make the problem worse. Since the problem goes away at full power (when everything in the cavity would normally be at maximum temperature), I would suggest you first try to increase the temperature of the KTP. If your laser doesn't have separate TECs for heating the KTP and cooling the diode, then try backing off on the TEC drive that cools the diode just a bit to increase the temperature that way.
Adam