How do I deal with an disloyal cabinet?

If you fire the other ministers one by one, it does indeed lead to the loyalty of those you replace falling. This is why you use the "Reorganize cabinet" mechanic, which fires all of them simultaneously and causes a much smaller drop in loyalty. Then you can replace them all and reinstate the ones that are appropriate. You also need to be willing to put ministers with very low xp but very high loyalty into positions when they have good sympathies for your current ideals.