Applied too soon?

Hi everyone! Sorry if this is a dumb question or problem. But I recently applied to be a cca, something I've honestly always wanted to do. I had been told the process would probably take around three months, a month to even hear back in the first place! And I'm doing a class that lasts three months so I thought I might as well apply now so I could have the job stuff starting around the time I get out. But now it's been less than a week and I've already gotten a job offer. I haven't heard from an actual person I could talk to, and I know background could still take a while, but I guess I just want to know what you guys think. Should I just decline the offer and wait until my class is finished to try again? Or should I go forward in the application and explain my circumstances, hoping they're ok with me starting in June? I'd love to have this job and feel bad for double commiting. Any help or assurance would be greatly appreciated! Thank you!