OK. I hope you can see what David was teaching you. Crops from the first row show what happened to your shadows as you moved the fill light in. The second row shows a crop from the upper left corner of the background. The camera chose an f-stop of 4.0 for you, that's OK. However, as you moved the fill light in the camera went for a quicker shutter speed 1/80 of a second vs 1/30 in the first two. So it got the white bear right and started dimming the background. Faster shutter speed = less light captured. So after looking through your camera manual, there is no way I see to lock f-stop and shutter speed - no truly manual mode. I think this does indicate that for this purpose you do need to consider a different camera. (I would recommend an entry level dSLR, or a good used older model dSLR). That is assuming that budget is a consideration.
If you are able to document a non-profit tax status and provide a few references of your rescue work, I may have a tip on keeping your shelter's cost down. Or if this is also for personal use I have my opinion on models just like many others here on the board.
These photos do look much better. But you will only be able to go so far with this camera. My opinion, others are welcome to chime in.