As
@gondar1 said, it makes sense to get in the cue
for a proper diagnosis but sounds theoretically like sciatica pain. Does it ever run into your glutes and down the legs or hips to legs? It can be debilitating.
Few things you can do that I have incorporated into every week's work - already mentioned reverse hyper extensions. I do not swing on the way down and control the downward movement, but I have watched a lot of videos where people swing. It is great for the spine or is for me. Be prepared to be humbled if you have never done reverse hyper extensions and start with low weight. I also do traditional hyper extensions and unless you like doing endless reps, it won't be long before you are holding plates while you do them for reps.
Great for lower back, glutes, hams, and stabilizing muscles. Generally two great exercises for your posterior chain.
Having a strong core is essential to keeping your back in the proper position, which I am sure you know. Vacuums are ignored a lot, but they are one of the exercises you can do at home and hit specific muscles to help stabilize the spine and keep proper posture. Not to mention they work specific muscles I find hard to hit. I am also a fan of the old fashioned mini crunch. You have to do a lot of reps, but you can use a piece of equipment on a mat that keeps you stay in perfect alignment and supports your neck (what ever the circle piece is that weighs about 5 lbs. I use it, but do not put my hands on it, to take away from the abs. I use one finger to ensure the pad stays on my neck.
One exercise I do everyday before I start any workout due to past back injuries and even on off days - 20-30 simple pelvic tilts. There is info says that it may be bad for sciatica but I find it has helped every kind of lower back/mid back pain. On your back, lift your glutes up like your woman is on top of you, tilt your pelvis, and squeeze your perineum and hold for a few seconds. Sounds odd, but it was a neurological osteopath that introduced this to me prior and it works for me. My sports rehabilitation specialist also had be doing this when I strained my lower back and it was heading into my glutes.
Simple hangs as long as you can, or even use straps and hang from a chin-up bar longer. Decompression and it is a great way to also help strengthen all connective tissue for multiple joints.
None of these exercises replace a proper diagnosis, massage, chiropractic adjustments, etc.
Best of luck getting this sorted out. It is pretty hard to do anything with a very sore back.