Sparklekitty is right about this of course, that self-esteem, self-worth is oh so important.
I know I did this to some degree when I was say, in my 20s. I probably still do it in some ways now, however I have curbed it dramatically.
This is just from self observation. Never really knew why I felt the need to always do the right thing as in whatever someone else thought was right! But often I went that way. Not any more. Now I make decisions on what is good for me (also for others to some extent, I'm not selfish).
For instance with that paper you got from your former employer, I would have torn it up and thrown it in the garbage. You don't work for them anymore. If they want to pay you say, 10k to sign the agreement, well maybe lol - short of that, why on earth would you do that?
As for making amends to them, you realize the way you quit wasn't perhaps a good way so you (i'm assuming) apologized. Well that's the end of that. You cannot control someone's feelings about what you do.
So you make the best decisions for yourself, you get comfortable with that and leave the other people to their own devices.
Honestly, most people are focused on themselves and what they want. You have left that company (for instance) you are a commodity to them, you put in your person hours for pay to work with them. That is over, forget them.
I know that is really hard to do, it's not easy to change habits (and that is what this is) of a lifetime. But this is as good a place to start as any.
Therapy, if you are interested in really knocking this on the head, would be a great thing.