This sounds pretty typical for a soft tissue injury.
Have you tried a good massage therapist yet? I've been to two over the years that I felt were amazing--1)was a chiropractor that did massage on the side and 2)was a former teacher at a massage school. Both were employees at different Massage Envy locations which had about 20 massage therapists on staff, and these were considered the ones to go to for actual medical issues. I'm not a fan of chiropractor's doing adjustments, btw...
I was in a car accident many years ago and was wearing the top part of my seat belt, but not the lower since it wasn't automatic. My lower body slid forward and I severely strained that same part of my lower back you're describing. After a year in physical therapy, along with various meds, I found a new pt that helped get me back to functioning level within a few weeks with a combo of stretching and massage.
Many years later I ended up with various soft tissue issues in my back that will likely be ongoing due to a rare neuropathic pain disorder--my back isn't my biggest health issue, but when it flairs, it can be miserable.

I continue to use the stretching technique that the pt taught me from when I was younger, along with a whole host of strategies.
Try taking hot baths with Epsom Salts twice a day. Then learn how to stretch out your lower back with a good physical therapist. I was taught to put one leg up on a surface about half way between hip length and knee length, and to step back as far as possible and then lean forward to stretch out my lower back. I'm not sure this is the best explanation...so please ask a pt. I would go for a massage at least 3 times per week. Next, try a very gentle low impact yoga class or personal instruction. (Tai Chi might be another alternative.) If you can find an infrared sauna, go as often as possible, preferably before the massage. Also, if you can swim in a warm pool, do so as often as possible. I'd try foregoing the ice during this period.
The pain meds will only mask the problem, so try and use as sparingly as possible. If your doctor is willing to start you on a course of steroids, that might be a good idea. I do a couple a year as needed. When I've had spasms over the years, my doc has given me trigger point injections, and they were a tremendous help, to calm everything down!
Consider switching to a diet that will not be as likely to cause swelling. I've found that the Paleo method helps me--check out the free info in the forums at Marksdailyapple.com for example. It will likely be completely different than what you currently eat, but I have chronic swelling and food does make a major difference.
Hopefully, you have a soft tissue injury that can be resolved quickly! If you do need more intensive intervention, get a second and third opinion before attempting surgery--and do them in a different city so that the docs aren't cronies that will rubber stamp what the other says.
Good luck.