This is a good post.
I would add, it's very common for the person in Abe's position to have an instinct to try to hide/minimize the interaction with the ex from the current love. It's a natural instinct because, as someone else said, there is a natural jealousy/competition between the ex and current one. But as soon as you become the slightest bit evasive, you start FEEDING the insecurity of your current partner. You're trying to humor your ex, but you're hiding that from your current and your current feels it.
One thing you can do from here on out is to try to include your fiance in any interactions with your ex. You need to drop off or pick up the child? Invite your fiance to join you, use it as an opportunity to take her out for a quick bite/treat/walk in the park just the two of you before the exchange, or the two of you and your child together afterward.
Even if she doesn't want to go, the invitation will make her feel less insecure that you're having personal time with the ex.
You need to talk with your ex about your child's grades, medical condition, summer camp, whatever -- talk with your fiance about it first. Any time you talk with your ex, mention it to your fiance. And try to avoid it when not necessary. Think of it this way - if you didn't have a child togehter, there would be NO reason for you to be visiting her to take her booze or just talk or whatever when you're engaged to someone else. Anything you wouldn't want your fiance talking about or doing with her ex's should be off limits.
Also, be honest with yourself about how much of your unnecessary niceness is really about custody. I'm not saying it's not, but the things you've done are very typical for men in your situation (as is your fiance's reaction typical for women in her situation). And the real reason most men do it are 1) to keep the peace/avoid conflict, and 2) Because it feels good to have your ex regret losing you/still need you. That's human nature. But indulging it will ruin your current relationship.
Buying the ex booze was definitely a bad idea, IMO. @Abe
, I'd still like an example of something flirty that you said to your ex.
Abe, for me the issue is around your privacy walls.
Imagine that a married (or engaged) couple builds a wall around the two of them. The two of you inside, everyone else - outside. Inside that wall, you have secrets that are just between the two of you. Outside that wall you can have other walls appropriate for family and friends. But, when you interact with anyone else, you NEVER build a privacy wall that excludes your spouse from your relationship with that other person i.e. no secrets that exclude your spouse/fiance.
I don't know what you said to your ex. Maybe it wasn't even that flirtatious. But, whilst you may not have overtly "hidden" it from your fiancee, you didn't openly share it with her either. I haven't had the stress of being in a second marriage with managing my kids and ex from my first; but if I was in that situation, I imagine what I'd want to be certain of is that my fiance's first loyalty lay with me and vice versa.
I'd suggest that you first discuss future interactions with your ex, with your fiance. Don't just keep her in the loop, but actively engage her in how you are managing your relationship with your ex right for the start. It probably isn't always practical, but where possible, ask her opinion on important messages you send to your ex before you send them. You're not asking her permission, but you are giving her insight into what (and more importantly why) you're saying whatever it is you're saying. You're also giving her the security of knowing that she can never be surprised by anything that your ex may claim you said/did.
It may be necessary for you to seem "nicer" (although this should have limits) towards your ex than you actually feel. If your fiance knows firstly, that you're doing it and secondly, why you're doing it, then (if I were her) I'd feel completely secure in your priorities.