Hi, Shelby:
There is no tool for automatically shifting to the right as you are looking for. Notation Software products are "bar oriented" so that notes are specific to a beat position within a measure unless moved.
Enter Sequential Notes ("sn" key shortcut) will change the length of existing notes to allow insertion of new ones and altering the length of succeeding notes within a bar to compensate.
Cut Note Duration ("cut" key shortcut) will cut the length of existing notes and leave rests to the right.
Neither of those are tools I've used a lot. I've been using Notation Composer so long (since before it WAS Notation Composer!) that I automatically think measure to measure and rarely come to a situation where I want to push anything over a barline. Your mileage will vary, of course.
I hope that helps.
David