Unemployment Question

I currently work part time for a college in a student support role and get paid pretty well per hour. However, my contract runs out in June and will be not be renewed as another full-time position will be absorbing my duties. However, the college has offered me another job (fulltime but way below my pay level and area of expertise), if I don't take the job (which I really don't see myself doing) can I claim unemployment or since there is a job avaible for me does that leave out in the cold? I am of course looking for another job in the meantime but just in case.

First post but thanks for your help in advance.