The fix for getting reliable seams and seam intersections (without going to paper piecing, where intersections will be perfect) is very, very careful pinning.
If you pin with very fine quilt pins (I like Clovers EXTRA Fine Quilting Pins...you can even sew over them) on either side of your intersection, the layers will not shift as they run under the presser foot.
Note that it will take some time and practice to get your method "down", I like to push a pin through the point to be matched first, at right angles to the fabric, before I place the pins on either side of the intersection. But you will get very good at it.
A great way to refine your skill on points is to make a whack of HST's (I love using the Triangulations sheets you can print out on your printer...fast and accurate for making a ton of HST's all at once,
https://www.bearpawproductions.com/s...-x54150408.htm), and then make a whole buncha pinwheel blocks. Assembling a bunch of those together gives you a great, quick way to practice points, and pinwheels work in all kinds of quilts, so no waste.