I sleep side-ways and takes cotton pillow but I don't have this kind of problems.Don't develop wrinkles on my face no matter if the pillow is pressed hard against my face.
Maybe the kind of pillow you are suing is too hard and also old pillows cotton breaks up and they are harmful with sleeping with them.
Get a new pillow which are soft and get from a company not from local shop as local shops use recycled cotton in machines to make new pillows which is not effective and breaks up in 2 or 3 years.