I usually win the fair play award. Then again my tactics are stay on feet and try not to slide so I don't get a lot of yellows. Most of the time I go without getting a single yellow. There are times where I will pick up like 4 or 5 though. Very seldom though.
I've played in the Welsh Premier League, and League of Ireland Premier. If you have "aggressive" defensive tactics, you are going to pick up some penalties, cards, etc. The game is testing you. If you adjust each time, I don't think it will get better. Hold the line. If a player gets a red, I usually issue a warning first time, fine second time, they should accept this. (That said, recently, when my player gets a red, I do not have these options - I'm not sure why - I don't see anything).
It gets better. Your players will learn. When my player picks up a yellow, generally, I tell them to ease off tackles, by going to Tactics -> Player -> Add the player individually, and tell them.
In a past year, I was not getting any penalty calls in my favor. I was, generally, supporting the refs. Once I complained about a bad call, and that seemed to restore the balance to some extent. I had supported the refs for years.