If you can do super heavy partial squats your core might not be the issue. It could also be a mobility issue (thoracic spine and hip flexors) that prevent you from maintaining a good posture in the bottom position of a full squat. Almost like your own body is pulling you forward. So you not only fight the weight, but your own body.
A mobility routine for the thoracic spine and hip flexors might do you more good than a core program (although the core is always something that can get stronger) and also doing plenty of front squats.