Burning nipple pain

Has anyone experienced burning nipple pain between feedings? My LO is 5 weeks, and I’ve been having burning nipple pain between feedings since he was born. At first, I had some cracking and bleeding so the LC said the discomfort would go away when that healed. Those have long since healed, but the discomfort between feedings has gotten worse. The initial latch is very painful on one side, but that quickly goes away. The LC suggested cold gel patches, but anything cold feels like daggers. Any suggestions or similar experiences?