I've struggled with the same problem a good while ago (training for over 4y now in my 40s).
I'd reccomedations looking into the Athlean-X YouTube channel, it's got kind of a duchy body builder vibe to it at times (which I'm not) but it's run by an athlete trainer and physical therapist who is excellent and has helped me very much.
Here are some of my top reccomedations:
- Warm up before BJJ and preferably before most of the mobility exercises/stretches you do.
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Hanging, passive (just activating grips, relax the rest) feet relaxed touching the floor behind you. I now do this every morning, worked wonders for shoulders and back.
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Put your back against a wall, arms our, fingers pointing up. Drag the thumb side of your hand or back of your hand up and down the wall, down from shoulder level and up as high as you can, trying to trace a straight line such that your elbow is bent 90deg along the path. There is a video on this on the Athlean-X channel.
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The chicken wing shoulder stretch. This one feels super effective but my results have not always been great.
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Use a bar (like a broom handle for example), grab a wide grip and use it to stretch, moving arms up and down, in front and behind your back. I'm sure there are good videos on this too.
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Sit on the floor with slightly bent legs in front of you. Place your palms down behind you with straight arms and rotating your shoulders out. Work on scooting your butt forward while keeping your back as straight as you can (your arms are gonna be further and further behind you giving you a stretch.
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Look into the book "becoming a Supple Leopard", it has some excellent stuff on posture and mobility.
I have no affiliation with either of the reccomended sources I mention, they are just among the things that have helped me with similar issues.
Oh and Yoga of course (see my post on yoga for BJJ).