Hi Pam, & welcome to the Board. Quilting on applique can add some really nice details to the front while creating very interesting quilting on the back of your quilt. It is also a good way to hide puckering or minor mistakes. All of that being said, may I ask what kind of fusible you are using that is still so sticky when you get to the quilting? If it were me, I wouldn't continue to use anything that is going to gum up my needle. You will end up spending way more time than you should cleaning the needle & hoping you aren't developing birds nests on the back (if some of the sticky stuff happens to catch a thread at the backing). I don't use much fusible, so I will not recommend a brand or type, but I'm sure many here can offer their experiences as a good reference to you.
I know my comments are not very helpful now that you are already to the quilting stage, but there will be other quilts in your future to consider. For me, the quilting is the most fun stage of the process; the part that can really bring the quilt to life. It should, at the very least, be just as much fun as the piecing, and not be something you dread because there is so much extra work involved. Besides, machine quilters are supposed to minimize their starts & stops to reduce the risk of things like jagged looking lines and curves. So, you certainly do not want your needle gumming up at the wrong spot causing you to have to rip out an entire line of quilting. Not Fun!!
Sorry I got long winded. I wish you all the best of luck with this and all future quilts, & by all means, please post a picture when it is "completed". :D