How to treat horrible dry sore throat, but no fever, cough, or sneezing and mild mucus drainage in the back of throat. I can barely swallow saliva.?

See your doctor. It could be that you need a humidifier and that gargling with warm salty water (once in the morning and a night) might help with the soreness. But, if you are having a difficult time swallowing, it is likely a good time to visit your doctor so that she or he can look directly at your throat to make sure that there is nothing serious interfering with you ability to swallow.
Dry throat . many things can cause a dry throat. First look at any meds or supplements. Next look at humidity - you are in a cold area in the winter. Heating systems take moisture out of the air. Get a humidifier. Finally, your state of anxiety/stress can cause a dry mouth - as well as your state of hydration. Drink plenty of water and use tartness (lemon juice, lemon drops) to stimulate your salivary.
Pharyngitis. Try salt water gargles, antihistamines, sore throat sprays. If no improvement see your doctor there are some oral infections that may cause such symptoms. .