If the fabric is very annoying - and some are - I would use a different one.
If you are going to go through all this work - use a good/better quality fabric.
I am one of those that wash everything washable before cutting it. Washing might make the fabric behave better.
As far as the wrinkles being noticeable after the project is completed? I think most people would not notice one way or another. But - again - if the fabric is not pleasant to work with, find something else to use.
(Okay - I admit it - I am older and getting to the point of not wanting -or needing to - fight with uncooperative fabrics.)