I've dealt with bouts of severe back pain from sitting at a desk for far too long through my 20s (and probably initially caused from a dumb injury lifting weights). I suffered with "throwing my back out" for maybe 10 years. It got to the point where I sneezed one time and threw out my back so badly I was bedridden for a full week.
But now I'm 42 and pain-free. Pilates (with a GOOD instructor) was what initially helped me. I started when I was still hurting badly and a good instructor can help you strengthen muscles and loosen what needs stretching without damaging you further. Mine was an active dancer and knew a ton about anatomy and injury recovery. I believe Pilates was originally created to help dancers recover from injury so it is a good place to start. Without knowing your underlying issue, most back pain people have comes from sitting too much, which means you can probably solve the issue completely through exercise and stretching.
Some small details that might help right away: a big key for me was learning how to "activate my core" which is something like sucking in your stomach but more conceptually like trying to draw your stomach back toward your spine. I believe it is a key element of many martial arts too, and by learning to tighten those broad, internal muscles you take stress off of the fiddly little muscles that are all knotted up and causing you pain. The more I was able to be aware of those muscles and activate them when my back was tight/tense, the easier I could loosen the knot, so to speak.
From your description of your pain, there is one key stretch that should address it directly. You might have heard this before from a doctor or P.T., but I highly recommend you do it routinely. Find a doorway and hold your arm up like for a high five. Put your raised arm against the wall with your shoulder at the doorway gently push your body through the doorway so you arm is stretched back at the shoulder. This should be a great stretch for your upper pecs and will probably feel quite tender. What is happening is that by sitting at a desk you are hunching forward which is stretching your back muscles and compressing your pecs. Exercising your stretched out and painful back won't do much, what you need to do is stretch your compressed pecs so they can take the load off your overtaxed back muscles.
But now I'm 42 and pain-free. Pilates (with a GOOD instructor) was what initially helped me. I started when I was still hurting badly and a good instructor can help you strengthen muscles and loosen what needs stretching without damaging you further. Mine was an active dancer and knew a ton about anatomy and injury recovery. I believe Pilates was originally created to help dancers recover from injury so it is a good place to start. Without knowing your underlying issue, most back pain people have comes from sitting too much, which means you can probably solve the issue completely through exercise and stretching.
Some small details that might help right away: a big key for me was learning how to "activate my core" which is something like sucking in your stomach but more conceptually like trying to draw your stomach back toward your spine. I believe it is a key element of many martial arts too, and by learning to tighten those broad, internal muscles you take stress off of the fiddly little muscles that are all knotted up and causing you pain. The more I was able to be aware of those muscles and activate them when my back was tight/tense, the easier I could loosen the knot, so to speak.
From your description of your pain, there is one key stretch that should address it directly. You might have heard this before from a doctor or P.T., but I highly recommend you do it routinely. Find a doorway and hold your arm up like for a high five. Put your raised arm against the wall with your shoulder at the doorway gently push your body through the doorway so you arm is stretched back at the shoulder. This should be a great stretch for your upper pecs and will probably feel quite tender. What is happening is that by sitting at a desk you are hunching forward which is stretching your back muscles and compressing your pecs. Exercising your stretched out and painful back won't do much, what you need to do is stretch your compressed pecs so they can take the load off your overtaxed back muscles.