What allergy medication should I take to avoid dry mouth?  

Newer generation. Benadryl (diphenhydramine) is going to cause the most issues with dry mouth. Newer antihistamines (allegra, zyrtec, and even claritin) are going to cause less. Anthistamine nasal sprays (astelin, astepro, pantanase) are the best option (rx only) to avoid systemic side effects (drowsiness, dry mouth, etc.).
None. Dry mouth is not likely to be due to allergies. Furthermore, most allergy medications can actually cause dry mouth as a side effect.