Try running. It's one of the rare scientifically proven ways to help heal disk problems. It's helped my SO and myself.
But not a lot. 1 to 3 km every day is fine. Walking 1/2 of it is fine. Just go out for 30 min and try to run as much as you can, but go slowly. Just a bit over walking pace is better than short bursts. You really don't need a lot and isn't any better for extreme runners. It'll get better in a week or two and it will start hurting again if you stop for too long before your fully healed (months probably)
It's all about pulsating vertical compression. And even if it does not help due to that, it surely will for other things.
I've been guided to instead cycle. If your back pain is caused by muscle imbalance (e.g., another leg longer than the other) or herniated disc, then running will supposedly make it worse. Yet, activity is important, which might be why running works for you.
The advice is very specifically for mild disk problem. Also for light running. You're not going to run any marathons. It's just a little more than walking. Overdoing anything will hurt you.
I am a runner, I had slight lower back pain before running that went away in the early days of running. I mostly assumed it was losing weight, which I am sure helped, but there is no doubt both my focus on posture and actual posture improved. So it is a bit of a positive feedback cycle.
To your point about no to much running, after I began running for periods over 2 hours, I would get some back pain during the run and stiffness after. However, I began doing pushups and planks, sure enough I strengthened my core and I don't get backpain on even 4+ hour runs (not to mention I developed a 6/8 pack which I never had in my life, a nice surprise in the mid-30's).
I had lower back pain for over 2 years. 7months ago I started running. Once every week for 1hour(~10km). First 3-4 runs I took pain killer, because I was ok during a run, but wasn't sure what will happen next day, then I stopped taking all pills and just went for 1-2hours run once a week. After a month or so the lower back pain was almost gone and I run more than 400km since. Of course, my whole body feels significantly better.
edit: You should consult that you wanna start running/swimming with your doctor!
I used to run when I was still losing a lot of weight. I'd still be running except that it seemed to cause an endless series of injuries and pains that no one could tell me how to fix.
But not a lot. 1 to 3 km every day is fine. Walking 1/2 of it is fine. Just go out for 30 min and try to run as much as you can, but go slowly. Just a bit over walking pace is better than short bursts. You really don't need a lot and isn't any better for extreme runners. It'll get better in a week or two and it will start hurting again if you stop for too long before your fully healed (months probably)
It's all about pulsating vertical compression. And even if it does not help due to that, it surely will for other things.
Good luck! I hope it helps.