Although they have similar definitions, sensitive and sentimental have distinct meanings in this context.
Someone who is sensitive is, as you mention, easily offended or upset, or is frequently emotional.
Sheli is a sensitive person and can't even watch movies with violence, as she is too upset by scenes of people being injured or killed.
Someone who is sentimental is overly attached to a memory, emotion, or context, and prone to nostalgia.
My mother is particularly sentimental and never threw away any of the birthday cards my brothers and I made her when we were kids.
For a position that requires dealing with things like angry customers, tight deadlines, or difficult subjects, you can say that someone is too sensitive for the job. For a position that requires getting rid of memorabilia or objects that may evoke nostalgia, you can say that person is too sentimental for the job.
Either works, but each means something different.