You can use the AdvancedArcMapSettings.exe utility to change the Data Frame refresh rate. But that changes it for all projects and since you have some that work and some that don't I'll guess that won't help.
Other than that I think you need to compare your projects and/or your annotation layers and track down the key difference if any. Can you move an annotation layer from a "bad" project into a "good" one and have it refresh the way you want? Or vice versa? If so look for differences in the projects or annotations. Anything that can slow the redraw could be a source of problems. I know that if labeling is on, it can really slow down redraws and seemingly freeze the screen in the process. Some layers are so complex, or their symbols are, that redraws seem to hang. Same for the annotation layer itself.
There are many other performance affecting things that may be present in one project and not another. So Google on ArcGIS performance and look for other possible bottlenecks that may be in one project or another. I think a typical ESRI suggestion would be to do your editing in a fresh project with as few other layers in it as possible, preferably without labels, complex symbology or definitions. If you
One last thought. If you are editing through the layout you have to redraw a very large area. It has the advantage of showing you just how the annotation would look in the final map but if you have a lot of things slowing redraw in a project you just might have to forego it, or you might turn off a bunch of projects, suspend labeling, etc.. Or you might find it much faster to go into the view to edit but do so at the scale of display so things look like they will on export.
