forgot, one thing about the dremel, if you use the fiber cut off wheels and do it free hand WEAR SAFTEY GLASSES!
I don't know if its the wheel or the wood, either way its throwing it in your face. I'd hold the dremel to the shaft, mark a line for cut off. This part is harder to explain than do, I'd slowly turn the shaft than plunge the bit to finish the cut. I do cut just shy of the line when I free handed, and cleaned up on a disc sander.
Sounds like its a lot of work, its a few seconds in entirety. Before I'd taper I'd make sure the tip of the cut off end was 90 to the shaft.
I used the disc sander to create a lip or shoulder on the point taper. You get this right and you'll never taper any other way! get it wrong and it will screw with your sanity! Best method is to have 2 wheels if you do a lot of shafts..one for 11/32 one for 23/64 and leave them alone! For nocks I used a jig on my big disc sander. Yup that makes 3 wheels (2 woodchucks and one disc). You can use one woodchuck but I found setup is a royal pain when trying to get the shoulders and swap shaft sizes. I never did use my woodchuck for nock tapers. the big disc worked great.
If that tips not perfectly flat, when using the woodchuck it will bind on the stop pin as you turn the shaft...or can. If you're experiencing binding, the first thing I'd do is bump the end of the shaft on a disc so its perfectly 90. I found the tips would ramp up (bind) on the stop pin, made for inconsistent tapers on the shoulders.