Staff Position - Teaching classes without contract or overload, offered another job. What to do?

Hello All,

(Hypothetical, I might find myself in this position soon) I am an academic advisor for a medium sized regional university. I am slated to teach 4 credits (4 one credit classes) this semester. However, I am not teaching these courses on an overload contract and I am teaching them on my advising contract time. Meaning no overload pay etc. I have another job offer pending, one that I most likely intend to take.

I am feeling a lot of guilt about taking a new position at this point when I am already committed to teaching these courses. However, the new position wont allow me to wait to start at the end of Fall term. What is the best way to approach my department and how much notice is appropriate?