I am slowly back to exercising. My story is complicated and unique (like each one here) so I will jump to the question:

Any interesting sports with very low risk of concussion or head bumps?
I used to enjoy tennis a lot. Not sure it is safe, I remember accidentally once hitting myself with my racket, and worried about whether a tennis ball hits my head (again likelihood is low and I usually play with girls - so their strikes are not that strong).

I am looking for some type of competitive sport that helps me socialize even if it is with just one other person.

Any thoughts? (other than chess )

I play golf personally. To make it more exercise intensive I just walk instead of using a golf cart. I also just casually play basketball. So i'll play games of pig or around the world with friends/family members, or put in earbuds and just go shoot around on my own.

Bowling, Table tennis, Swimming

I wouldn't worry about tennis. I play ping pong. Chasing a frisbee in any of the many permutations of frisbee games would be safe. Shooting baskets is OK but we have had many who have collided heads on the court playing man on man.

Soccer without any heading can be OK if everybody agrees.

Swimming and weight training. And running if it doesn't produce symptoms?? Swimming isn't exactly social but I used to go with a group that had a social/potluck every month. I wear dark goggles to help cope with the light (even indoors) and shadows. I have neck damage from my accident, so I switch to a snorkel when my neck starts to tire/ache. I swim mostly in the pool now - the motion of the water and my head can make me dizzy, but I'm a strong swimmer and confident that I'll be ok swimming in a pool that is only 3.5 - 4 feet deep.

Weight training also isn't necessarily social, but if you join a group or a class, it can become social. Just make sure the instructors are qualified and won't push you to do anything harmful. I like some of the Crossfit type workouts, but I've also seen some crappy technique and people being encouraged to do crazy (stupid) things - the last thing we need is another injury!

Another thought to help get you started is something like a fitness trail that you can do with a friend or two - you can walk or jog and then do the workout at each exercise station with any modifications that you need.

If you're looking for competition, you can join a master's swim group and attend meets; try some local 5k's; record your progress with weight training or check out a crossfit "box" and do some of their competitions; or track your progress on a fitness trail - you can work to go faster, or do more of the exercises, etc.

Good luck, be smart about it!
Diane