Take another look at the original sentence:

Sherian refuses to let her kids have a puppy, she is convinced that dogs are just big germs and refuses to follow one around with a sponge and bottle of bleach.

You chose to fix it the correct way:

Sherian refuses to let her kids have a puppy, for she is convinced that dogs are just big germs and refuses to follow one around with a sponge and bottle of bleach.

Sherian refuses to let her kids have a puppy is a main clause. She is convinced is the second main clause. A comma and a coordinating conjunction will always fix a comma splice. Since for means the same as because, it makes an excellent connector in the example above.

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