I think you should try the A7ii with your old glass and then go from there if you want some new glass this is a great camera for the money and it is on sale now.
Put a lens on, put in on program and start shooting. In no time you will like it and begin using other features.
if I WANTED ONLY ONE LENS, I WOULD TRY THE TAMRON 16-300.
Think about and look at a Sony A58. It is on le for $399.00 and can be used on program or you can get into more control of what you shoot.
With the little kid moving, try shutter speed for a while.
The Sony A58 is on sale with a kit lens for $399. If it feels ok, it would be a good choice.
I know when I no longer want to shoot. then I work on post processing for a while and then go back to see if I can shoot better and have less post processing to do.
I shoot Sony, but since you are familiar with the Nikon I think you should stay with this brand.
If you do not have any extra Nikon lenses I think that you should get the Sony. Mirrorless is the wave of the future.
This depends on your budget and skill level and how much new technology means to you.
think you should start with a Sony A3000 or A58. They will both be great at action and low light and do not cost too much.