I'm pretty much the same. Sixty and have been racing since I was 13. I'll running anything from the USATF Masters Champs to a local 5k. I still train systematically, and aim to run the best times I can. If I go to a major 5k I'm going to be 3 minutes behind the winner; in a smaller race, up until about 3 or 4 years ago, I might be battling a decent local high-schooler for the win.
I generally go to the race by myself - I'm married, but my wife doesn't want to get up at 6 pm to go see another 5k - and I generally arrive, warm-up, do the race, warm-down and leave. For me it's about the running, not socializing (I train by myself too - just a function of there not being too many other runners in the relatively small town I'm in). Like Barra, the training and racing gives me some structure and something to look forward too. I'm also not worried about what anyone thinks, although in my experience generally people of all ages respect someone who is older and keeping in shape, or who seems relatively fast (and 16:05 is relatively fast), as long as they don't act like a d*ck about it.