Taking action photos indoors without a flash.

That is the "Achilles heel" of the small digital camera
.. (meaning the weak point of the small digital camera)

A digital camera that can do what you want, will cost over $700.

Other solutions:

1. Use a flash, but have someone distract the puppy so that it is not looking at the camera. That should eliminate the "ghost" eyes.

2. Set-up one or two lamps with bright lights (about 200 watts each).

.........

New camera:

If you have an interest in learning more about photography, think about getting a camera that has some manual controls too.
The Canon A590IS is very affordable and also has image stabilization.
You can keep it in "auto" mode until you learn how to use the manual controls.

If your interest is toward a ultra compact camera (without manual controls).
I suggest you get one that also has an optical viewfinder so that you can use it in very bright sunlight.
Not many ultra small cameras have viewfinders.
Look at the Canon SD series of cameras with the 2.5" LCD:
.. SD1100IS, SD850IS, SD950IS, SD770IS
Look at the Sony W series of cameras:
.. W120, W130, W150, W170

...
..
.