I do very few exercises that directly work core muscles, though most of the weight training I do engages the core muscles to some extent.

I have had problems with my sacroiliac on the right side for years. Regular stretching has been the most important thing to manage it -- hamstrings, hip flexors, IT band, quads. Foam rolling my legs also helps.

BTW, even if your mattress is not worn out, it might not be the right mattress for you anymore.