I think you can use your EAD. Your H1 can't not be transferred because those are 2 different H1 types. The H1 sponsored by non-profit does not have the quota-limitation. The H1 quota for this year (for you to work in the company) was already used up. You have to wait for 4/1/2008 to apply for the H1 that will be effective on 10/2/2008.