If you are really interested with the second position, then give them a ring and let them know of your position now, let them know that you are really interested in taking up the role with their company etc and see what they say. Most will make the decision if they are really interested in offering you the job!
I think Dracoz provided you a very good option. It is worth to try it out.
If I were you, I would sign the offer first and keep looking for jobs I like. I would not be too picky when i were unemployed and would keep "upgrade" my payment and position after I start working.