I would check that Doug, with them. Last I heard they had a 100% rate of healthy pets. They don't just kill if there's no room.
MHS doesn't answer questions here. They will always take them. If they don't have room and it doesn't adopt, they euthanize. If that's a concern take it to a no kill shelter
How do you know she is spayed, if you found her? You should take your day to a cat rescue, so she has a chance for a new home.
The humane society has a program that helps with food. You might want to look in to
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