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 wanted to fix it this way:
Sherian will not allow her children to have a puppy,
but she is convinced that dogs are just big germs and refuses to
follow one around with a sponge and bottle of bleach.
Using a comma and a coordinating conjunction like but is a perfectly good way to join two main clauses. You must be careful, though. But shows contrast.
In the example above, no contrast exists, making but an illogical choice.
You might want to consult the rules for fixing comma splices and fused sentences.