So, it seems there's not an easy way. For now, I'll just copy the degree symbol so I can use ctrl-V to insert it. That seems to work when entering relative coordinates.
It would also be nice if TurboCAD were a bit more forgiving in this, allowing spaces also. Also, I second the motion to use something like N 10d5m32s E (with or without spaces). What's the process for getting this change on the docket? It worked similar to that in my old copy of QuickCAD, if I remember correctly.