Straight stitching can be used on appliques. If you are good at free motion, some people even free motion straight stitch their applique edges (avoids having to turn the applique around all the time). I would consider using a fine thread -- maybe silk? -- for straight stitching.
Also, check your fusible instructions. It should say whether your fusible is permanent without stitching. A few types of fusible don't require stitching. Some of those that are really permanent may gum up your needle if you also try to sew through them. So, for safety, check your fusible before you do anything. If you don't still have the packaging but know what the brand name and type was (Steam-a-Seam, WonderUnder, HeatnBond Lite, etc.), post here and someone will be able to tell you if it's the permanent type that shouldn't be sewn through or whether it's the type that benefits from sewing. A few work either way (I think Steam-a-Seam is one of those.)