Review the original item:
Sherian will not allow her children to 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 the correct way to fix the error:
Sherian will not allow her children to 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 will not allow her children to 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.