Do you pin your pieces before sewing? Sometimes I can sew long strips without them, but if the pieces have seams, I definitely pin.

I got a video called Sally Collins teaches you precision piecing and follow all her tips. It improved my piecing a lot.

Do you pin your projects? I was having the same problem because my pins-those yellow ones- were to thick. My teacher suggested pins and they really difference.

I will jump on board here about starching fabric before cutting. I pre-wash all my yardage, mostly to remove all the chemicals. Then I dry the fabric to barely damp. I use Sta-flo and mix 1:1 ratio or even 2:1 (sta-flo being 2) and give the fabric a good spray, back to wet. Then put back into the dryer until barely damp and then iron without steam. I don't worry of the starch flakes because those will slough off while working on the quilt and will wash out when the quilt is finished.
This makes fabric cutting so much easier and faster. The cuts come out very clean and your rotary cutter will last twice as long.

The good thing about fabric is that it stretches - you can nudge it a bit to make it fit. The bad thing about fabric is that it stretches - it makes it difficult to be consistent. We have to remember that we are working with a malleable medium which can shift and distort when cutting, sewing, and assembling. I have never been very successful cutting multiple layers of fabric so I don't do it. For others this is not a problem. Some can sew right along without pinning. I have to pin for things to come out right. The way you pin and the types of pins can make a difference. I have the agree the thinner the pins the more accurate my matching is. Some people can sew fast and chain piece. I find that I have to go slower and be careful about chain piecing because I lose track of which piece goes to what. Some people can finger press and move on. I find that I have better luck pressing with an iron before I move on. Some people pull their fabric through when sewing. I find I have better luck by letting the machine pull the fabric through as it stitches and only using my hands to gently guide the fabric through. The point is what works for some doesn't work for others. I think practice makes a difference and finding your own rhythm will come. Use the tips of others but don't become so OCD over it that you lose the enjoyment.

Pin every seam.

I nark with numerous layers of painter's tape to make sure I constantly sew an accurate 1/4 seams. I also like to trim my blocks down after press and starching. You just have to find the right combination of what will work best for you.

This video will show why your piecing is the reason your blocks won't turn out right: https://www.youtube.com/watch?v=aBoLPslLxTs

I was going to say cutting as well. slowing down really works well to. Lots of good suggestions here

Lots of good hints here -- now you just have to figure out which ones work best for you! Good luck. Quilting is a continual learning process.