I also cut my binding at 2" wide most of the time. Some times I cut the excess batting a little bit beyond the edge of the quilt when trimming it and let the extra roll into the binding to make it fuller. You can also take a deeper seam allowance to take up the extra binding width. I don't mind the binding being wider on the back side either.
The rhythm of hand sewing binding is so peaceful and usually the last time I get to pet the fabric and reflect on the quilt before it is gifted. It is another ritual that I love about quilting.
peace