Yes, professors are busy, but they are willing to write recommendations. (I was a college prof.) So, please don't hold off on that. You do need them. Now is not the time to be shy. Visit with them personally. I made the mistake of putting a recommendation in a prof's mailbox when he wasn't around, and he later claimed he didn't get it.

Your research plans sound good. At least that will give you something on a resume. Did you do lab work like what you want to do in your classes? At least you could talk about your lab experiences, if you had any. Otherwise, you could check how much experience these places expect and go from there.