I'd be surprised if your hand warmers generate enough BTU's to have much effect, a hair dryer or heat gun on low requires more time and effort on your part, but they will definitely get the job done. The trick is knowing exactly where the drain line is frozen so you can focus efforts on the problem spot. One thing to keep in mind if your drain pipe is PVC, the melting point of PVC is only about 180 degrees.
it is a bit unusual for the drain to freeze before the water lines. The best preventative for the future is to insulate both water and drain lines. If your typical winter weather doesn't justify the expense and effort, simply keep a pencil sized stream of water running when the risk of freeze-up is high. Too small of a stream of water may keep the inlet from freezing but not the drain, in which case you wind up with a real mess.