Thanks. It was very strange how my denial went down. I talked to her on the phone and she said "we denied you based on the income in the pay stub you uploaded, but I wanted to talk about your trip abroad...." I was more interested in what she had to say on the trip abroad as well, because I figured I could easily resolve the income thing later on. So I never asked about it. Then they told me I was ineligible based on being abroad.
But now that I've done my own research and I don't think there's a true reason to deny me based on me being abroad, I think if I appeal the "income" rejection, I'd be good.
I also made a new application and have been trying to get through on the phone line. It's impossible to get a hold of them. I'm giving it a week, then sending in the paperwork asking for the state hearing.
(It's also better to appeal the original decision rather than apply again, because it backdates to January and I don't want to pay the fine. Even though I don't plan on working this year, I want to to reserve the right to earn some income without being penalized for their mistake. On principle. Also I want to convert my 401k to a Roth.)