Mr. Tench, the English dentist, is a good example of someone who feels abandoned. He resents his position and feels that he is alone in the town, unable to hold a conversation with any educated people; unable to do his work at a reasonable professional level because he lacks proper equipment. He becomes, ultimately, rather barbaric in his resentment and lapses in his dedication to his craft.
Mr. Tench chose to come to Mexico, but he certainly feels abandoned. He is "without" - without the comforts of drink, conversation, family, etc. The whiskey priest, in constrast, is "within" - stuck in the state, stuck in a difficult role, penned in.
The priest drinks heavily, a symbol of his desire for escape, and he strays from the priestly regulations that dictate his actions (he fathers a child). Unlike Mr. Tench who reflects on a sense of being alone, the priest reflects on the pressures he suffers and comes to connect with those he encounters.
In the end, Mr. Tench still resides in the town. The priest escapes the state then returns and is killed. The pressures of his role/obligations were too strong to escape.