How big is the overhead compartment in a Delta plane?

Overhead compartments vary based on the plane (not the airline.) But your backpack should be fine. I used a bigger one on a few flights this year, and it fit fine in both commuters and regular jets. Sometimes I put it in the overhead, and sometimes I just put it by my feet, under the seat in front of me. It's a really good idea to do what you're doing- saves time waiting on luggage, and reduces the risk of losing it.