Thanks fuzzylogicians! As you said, I don't think it is a good idea to accept an unfunded offer.
Even though the school I got an offer with a full funding package is famous with computational linguistics, during the interview I found out I am not that interested in my POI' project. Also, I am waitlisted in UW computer linguistics program and I haven't heard anything from them yet.
I am working in the company related with computer linguisitics now, and I already told my supervisor I will quit because I start my phd this fall. ( I guess it was too early to tell him!!!)
As you said, I need to email my POI and ask him about the timeline. Thanks!