Your blocks are off center and off grain for the pieces that you are putting together, therefore each block is different. They need to be consistent and the same.
I've made several wedding ring quilts using pieced or templates and/or both. The 1st one I made gave me total fits because of how I cut my rings out on different grains, it was a "scrappy" and my seams were just totally inconsistent. Every block turned out how yours looks.
The fix is to cut your ring template fabrics out on grain (NOT on the bias) then cut your triangle template fabrics ON the bias.
The next trick is to PIN PIN PIN the whole piece in 1/2 increments from the center out to each edge. What I did next, was to find the CENTER of the triangle piece and the center of the ring piece and stitch FROM the center to the end of one side, then the center of the other side. Once I did this, the only "easing" I had to do was between 1/2" segments. This made a huge difference and made my pieces more accurate in the end.
I agree with you that cutting the triangles a lot larger, BUT ONLY ON THE OUTSIDE STRAIGHT EDGES will give you the fudge factor you may need for trimming to size before you sew the blocks together.
The other trick in sewing the blocks together is in the pinning technique the I learned from Eleanor Burns in her triangle work. That is where you put the pieces right sides together, then take a pin in the very point of where the seams meet at the template and the triangle seam to the other one - put the pin in straight down, then only "grab" about 1/8th of an inch. This one is hard for me to describe without being able to show you (I'm so visual).
Another tip with stitching the wedding ring is to make sure your pressing is consistent throughout the quilt so that you can "butt" your seams. For example, I would press the ring seams out towards the triangle fabrics. They will lay a lot flater. This is critical when working on anything with lots of seams that have to meet up (stars work is like this).
I hope this helps. I've made 5 wedding rings and am working on my 6th, which is a paper pieced version that I've decided is much easier and less brain damage. It's giving me the precision that I want. Even though I claim to be a recovering perfectionist, when it comes to the Wedding Ring pattern, I am a perfectionist through and through. Ha!