I'm ecstatic to report that Natalie has started to consistently hold on to a regular swing. She did this for the first time for any length of time at this park where the swings are very low to the ground (makes me less nervous if she falls). And once she showed me she can do it there, I tested her out on the big kid swing at home, and she's done great.
She has forgotten, and fallen a couple times at home, usually when she wants to sign something, but she immediately gets back on the swing and keeps going. I just tell her she has to speak when she wants to tell me something, instead of signing it!
Of course, we don't swing her very high when she's on a big kid swing. If she wants to "shoot the moon", she still has to be strapped in to an infant swing.