Dry mouth. Sore throat. What can it be? Mono. Strep throat?

Possible. Dry mouth is not a common symptom in either mono or strep throat, although, both can probably have people complain of that. Sore throat is common for both mono and strep but can also be seen in a lot of other viral illnesses as well non-infectious diseases such as allergy associated with postnasal drip, esophageal reflux, nervous spasm (globus). More detailed history is paramount in this case.
Impossible to. Speculate a diagnosis over the internet. Rec an evaluation by your physician for a proper diagnosis and appropriate treatment. Hope this helps.
Dry mouth. It may be neither. There are other conditions that can cause dry mouth, therefore it is important to see your doctor for evaluation and treatment.