mszabo is correct. The turn signal blinker speed is a factor of the resistance of the turn signal system. That is why when one of your lights burn out, the resistance changes and thus changing your turn signal blinker speed. Replacing the high resistance incandescent lights with LED lights
changed that resistance.
Grounds are also a factor or shorts in the wiring.
From your description, it may be that you have multiple problems with your electrical
To trouble shoot the problem(s), replace all of the LEDs temporarily with the old incandescent lights. Resolve your other problems one by one. Then put the LEDs back in. All you should be dealing with now will be the speed of the blinkers.
There may be a blinker module designed for LEDs which you can install to replace the old module. Check online with third party suppliers. I am sure you are not the only one with this issue. Also check with the service department at your local dealership. They may be able to give you some answers. Keep in mind they will give you the answers that suit their bottom line. Occasionally though, you will get an honest mechanic who will give you straight answers.
Remember "Where there is a will, there is a way". Nay sayers often push me to accomplish what I am told is impossible. That is why our country became a great nation. A lot of nay sayers and a lot of people proving them wrong.
Best wishes with resolving your electrical