I had the same experience - trained in a half-frozen garage, and at some point instead of just changing clothes, I decided to "harden myself" and rode home in a wet shirt. The result was predictable: two weeks on the couch. Then I figured out it wasn't about the cold itself, but how you deal with it.
Now I bring warm layers and put them on right after my workout, still in the gym - no hanging around in wet clothes in the cold. I don't forget a towel either, to dry off the sweat before changing my shirt. The key is not letting your body cool down in wet clothes - that's not hardening, that's messing with your immune system. If the gym is really freezing, you can even skip changing during the workout, but right after - no question, you go dry.
As for the gym itself: 12-14 degrees isn't critical for training as such - when your muscles are working, you won't feel the cold. If everything else about the gym works for you (equipment, vibe, convenience), there's no need to switch just because of temperature. But if the management has completely given up on heating and keeps promising they'll fix it "sometime," better not risk it and find a decent place. A cold gym is one thing, but combined with careless owners, you can't expect anything good to come from it.