than I would my own soul?

"I'd no more leave her in your care than I would my own soul."

Is this saying that "I won't leave her in your care like I won't let you take care of me?"

Yes, that's it. English is using an "elliptical" construction here which is very common. That's where words are omitted and repetition is avoided but the meaning is still retained.

Look at that statement this way:

"I'd no more leave her in your care than I would [leave] my own soul [in your care]."