The reason your spine hurts is because the hip flexors, typically used during sit up style exercises, are attached to your spine. Your core is hanging on for its dear life during extended sit up style exercise regimes in order to save your back.
Over 800 newtons of force are placed on the attachment points in the spine during sit ups. If you want to mitigate that, execute a Mcgill sit up-one leg extended, one leg bent.
Ensure that when executing a crunch, you only perform 30 degrees of flexion in the spine. Shoulder blades should leave the deck, but at no point should you be "sitting up".
I'm going to agree with others here in saying that there are many exercises that utilize the core anyway-if you're doing it right you don't really need to focus on the core.
I ensure that when prescribing, my clients do not have too much sit up style exercises. Planks are king, not sit ups. Planks don't involve spinal flexion.