Rather than splash out on a full frame camera, have you thought about the settings you are using in your OMD? For photos of moving subject you'll need some kind of continuous AF. Try using the centre focus point and setting your camera up with continuous AF and a high shutter speed (at least 1/250th if your daughter and /or the dog are moving).
You should get excellent shots, in sharp focus, assuming the OMD has half decent continuous AF abilities. If they really are moving fast, like running, then a traditional DSLR (ie with a mirror) will probably keep up better as they use phase detect focussing, rather than contrast detect. However you've said they are just pottering so your OMD would definitely do the job fine.
Even an enthusiast level compact camera can take sharp shots of moving subjects in good light if set up correctly with the right technique, so it's well worth reading your manual and trying different settings.
The D600 is an awesome camera - I've had mine for a couple of months and it takes the cleanest and most detailed images I've ever seen - but it may be overkill for your needs.
Hope this helps!
