I feel this same way as a teacher in a high school.

I teach resource room and literacy for learning challenged students and my students pop in on my all day long even when they are not assigned. That means during my prep periods as well, so there is never time to get any of the administrative or planning parts of the job done. But I love these kids and want to help them as much as possible, so how can I say no when they need a quick consult on a research paper or a difficult assignment or just to vent?

I think it's really nice that you have found a way to connect to T, by sharing a common experience.