Be cautious about this advice.
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).
edit: You should consult that you wanna start running/swimming with your doctor!
Dont wear it all the time - but it helps to remind yourself of your posture and put your awareness on your spine.
Wear around the lower back.
Now it won't go away no matter how much physical therapy I get or other exercises I do.
I think there was a meta review, but will have to ask my SO, it was her research that lead us to it.
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.