First off congrats on the newborn - just had one too 4 months ago to add to our existing one who's not-so-newborn anymore
For what it's worth, I'll mention that until 3 weeks ago I've been shooting with an XTi (400D) for 6 years.
Once the first kid came, and indoor shots became more prevalent in my photography, I saw the same issues as you and the same limitations. The single best thing that helped me was getting an external flash (580EXII) for those indoor shots with the kids. If you already have one, just disregard this for a moment, but if not, it will absolutely help. Just bouncing it off the ceiling was enough to make up for the limitations of the XTi.
As for T4i vs. 60D - the 60D seems to be the better choice except one thing to note is that the T4i is able to go 1 stop higher in ISO. This could be important for getting indoor shots that are in focus and motion-blur free if you're not using a flash since you'll need the higher ISO's to get fast enough shutter speeds.
However, I am not familiar with the high ISO performance of either camera. I would recommend to check out POTN (photography-on-the.net) which is a Canon fan-site for all things Canon DSLR. There are going to be tons of image samples from both cameras as well as discussions on ISO performance, etc.
One other thing to consider is that the 60D has a better viewfinder (much larger and easier to see through) than the T4i, which again once you experience it, it's amazing how much of a difference that can make.
So there's some give and take, but if the 1 stop higher ISO of the T4i isn't a big deal - the 60D goes up to 6400 (and 12800 on H) - the 60D is otherwise the better choice from what it looks like.