Take a look at the original sentence again:
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.