- Try and list people who work there, who are on your side, as references when applying.
- I know LinkedIn is hated, but in my experience, employers do check it. Ask someone who is on your side to write a brief blurb about working with you (it's a site feature, can't remember if it's called "reference" or something else).
- Maybe swallow your pride and try and smooth things over with the ex-employers by sending them a positive sounding email wishing them the best of luck, and mentioning how you enjoyed working with them. You might mention that it was constructive for you to leave since you felt like you needed to move on for personal reasons and it was for the best (a nice way of saying you disagreed with them but no hard feelings). Do not mention references/background checks or that you could need anything from them. However, do mention that if they need something from you, e.g. some explanation or document that they need, you might be able to assist them. If they aren't terrible people, they might view this as a satisfactory answer to their question of "what the hell was his problem?!" and give them something to rationalize what they perceived as you being cranky, and thus put the drama to rest ("he just needed to move on" or something to that effect).