Doc diagnosed my lower back pain to be caused by ligament strain but can ligament strain cause sacrum pain n a sensation down the leg as well?

Back Pain: Yes, actually it can. If the sacroiliac ligaments are strained there can be instability of the sacroiliac joint. This in turn can lead to irritation of the sciatic nerve positioned next to the joint. This could give sciatica as well as referred pain down the leg.

Back pain: The back pain can be caused by muscle strain, spinal stenosis, ruptured disc, nerve impingement ... It sounds like you might have a nerve pain. You should seek help from a health care provider, you might benefit from a comprehensive evaluation.