Sitting on a birthing ball opens up the pelvis and possibly destabilises SI joints further if they are already out of alignment so definitely don't recommend doing this.
I have 2 immediate suggestions for you:
1. Get yourself a Serola pelvic support belt online ASAP, it goes round the hips and pulls the pelvis together stabilising it. This comes highly recommended by osteopaths, chiropractors and professional physios for women with PGP (pelvic girdle pain) which is what you likely have. Use that on a daily basis to help you keep your pelvis supported.
2. Lie on your back (if you can), get someone to hold your knees together close (so that your knees remain hip width apart), push them open gently against this resistance (knees don't actually move apart, just push against the hands), then release hands, hold something between your legs (like a folded towel, same thickness so that they are again a few inches apart) and squeeze gently. This mavuever taught us by physios can put misaligned SI joints back into place and might help ease the pain. Go slow, try a little bit, don't overdo, just one repetition especially the first time. Let me know how you get on?
I'm a pilates therapist in training wanting to specialise in pre and postnatal fitness and I know a bit about PGP /SPD through my own experience and through training hence the above recommendations