Are there any home treatments for diarrhea and dry mouth?

See your PCP. There are many causes of dry mouth like side effect of certain medications, diarrhea, certain diseases and infections (diabetes, anemia), nerve damage, just to name a few. See your PCP for proper diagnosis and treatment or referral to a specialist.
Dehydration. If you have had excessive diarrhea, you may be dehydrated, thus the dry mouth. Can't tell what ails you over the internet.. A full medical history is necessary as well as lab tests. Best to be seen by a local physician to be treated for this problem. Failure to treat can land you in the hospital.
Source, not symptoms. Need to determine source, not treat the symptoms. Get to a Dr. and hopefully with evaluation and tests, you will discover the source of your ailments.