Also, there's more than just one tool for the job, because the job is constantly changing...
There might actually times that you're with the baby and don't have a full-sized camera in your hand ready to go. Don't forget to use your mobile phone. Get used to using it and figure out what angles work best. (For example, shooting from up high looking down will not always be a nice picture. Getting close and shooting at eye-level can be very effective.)
I have a 6D and a 5DMkIII (and a 60D, and T3i, etc.). Babies (and the circumstances around them) are constantly changing. The AF on the 5D3 is pretty awesome. When it comes to keeping up with the action, I've been frustrated with the 6D (original model) in anything but plenty of light. The center point (on the 6D) is fantastic. The (few) side points aren't as much.
Also, a FF camera is not an "upgrade" from a crop-sensor camera. It's a *different* camera. Even when lenses (i.e., EF mount) are compatible, they're really not the same lenses. They have a different "feel" and therefore will be used in different ways. If you're headed towards FF, then go for it. But FF will not improve your photography over a crop-sensor as much as getting better lenses. And getting lenses will not improve your photography as much as getting better skill and experience.
You don't have to upgrade your equipment to upgrade your photography skills. (Although it certainly doesn't hurt in many cases...)
Finally, the huge advantage of using your mobile phone camera is that photographing babies will inherently require all types of social media work. Posting this, that and the other. The mobile phone may be more helpful than you might think...