How should I go about applying for positions? I went through a nurse residency program but left right before I was alone on the floor. This has happened within the last few months, but should I try to apply for a residency or an RN position?

Go back to the residency program. You will be glad you did. It might seem like a waste of time now but eventually it will help you build some of the basics needed to be a great nurse. Not every residency is alike and some are way better than others. Hopefully the hospital provides a valuable experience from the residency they offer and you're also hopefully able to have a good experience.