Along with  HappyDance , I encourage you to consider each of your strengths.   If you're both performing the same task this duplicating work that is of no extra value.  Plus, it may actually be a drain on him to do that task while instead, you find it invigorating.  In our marriage, I do the details of the day to day budget maintenance, but the overall plan is one that my wife & I agree on.  There is momentum that comes from working on a goal together and each working within their strength zone for the good of the "team".  

 You're absolutely right that the budget should not be something imposed on him.   But it's OK to have different roles in the budgeting process. 

Ah, learned helplessness. Is this something that's an issue in other parts of your marriage? You don't have to answer that here, but maybe consider in what ways you've navigated similar issues successfully.

IMO, avoiding/counteracting learned helplessness is about setting boundaries. Teach, guide, answer questions, point to resources, but don't do for him. If he messes up his credit cards or something, help him figure out where it went wrong, but don't leap in to fix it for him.  It's a lot like coaching a kid through their math homework - you can't do the homework for them, or they don't learn. But you can point them in the direction of finding the answer for themselves. 

As far as keeping him engaged, can you make it part of your routine that you both reconcile your reconcile your registers over morning coffee, or something? Set some celebratory milestones like going X days without using his credit card, or reaching Y days AOM (or buffering), or hitting $100 in his e-fund?  Don't just save it for big milestones like a $1000 e-fund or paying off a credit card. :)

Do it in baby steps. It is hard to unlearn things. My ex-girlfriend was in the habit of lecturing about the budget, made the monthly budget meeting last way longer than then needed. I would say I trusted her, here are what I see we need. I would of been fine with that.
Maybe a short little class on how YNAB works, then let him work with it, then check back in with him after a few days, maybe show him a trick or two, tell him about the classes YNAB offers. You know your husband and hopefully his learning style and what he responds to, play to his and your strengths as a couple if that is how you work. 

My son is a budgeter and his wife is not. What they agreed on is he handles all the money and gives her X to run the house (groceries, kids needs, clothes, etc.). Could you manage his budget for him and give him X a week that he can spend on whatever he wants? Show him what his budgeted money is being used for. Maybe over time he will be more interested.

You have received lots of great ideas.  Just to add another one, how about working toward a shared goal?  Something you want to share and each has to put in 50%, something like a meal out, movie, coffee.  Then work toward larger goals, baby, own home, etc.  

Good luck to you.  He doesn't get that he needs to save more than he currently is (debit card refused so used Credit card), I know I did.  

Sea Green Gazelle said:
What I WANT to know is this: how do we dig out this hole without him losing motivation? And how do I avoid the temptation to just swoop in there and save the day?

Frankly, I'm mad at myself that I didn't push about him doing this sooner. And knowing him, he'll need a lot of handholding.

What is it that you really want, here? And what is it HE really wants, here?

Because it sounds like what you really want is for him to magically come around and decide he wants to do things your way, which probably isn't realistic? :)

I say this because I've been dealing with something similar with my partner - we both have debts and decent incomes, but we both have really different perspectives on money, so I've had to accept that our paths out of debt will be different. I'm having a lot of fun figuring out how to squeeze my own budget to start tackling my own debt (now that I've gotten a recent salary increase), and when he shares his budget with me, I can see so many places *I* would cut expenses and do things differently - but I've had to realize, I don't get to tell him what to do with his money, because like you, we are choosing to keep our money separate, and that's what that means. :)

If you want to control all the money and he agrees, then that would be one thing, but if you each have separate money, then you don't get to control how he handles his, as much as you might want to. You can offer suggestions and assistance if he wants it, but ultimately the choices are up to him.

So I'd encourage you to talk about this in a big-picture way, about your priorities and goals, and his priorities and goals. And then you can start figuring out how to help him work towards *his* priorities and goals, which will help you both figure out what will really motivate him.

Because otherwise, sure you could swoop in and pay off all his debt and "save the day" - but until he figures out for himself why & what he wants to change about his financial habits, nothing will really change, and he'll probably be back in the same debt boat again, sooner or later.

I'm not understanding the two extremes in your situations. Not that it's any of my business, but is this $10,000 of debt entirely discretionary spending on his part? Are we talking out of control and totally oblivious robo-spending? Or is he going into debt paying for a portion of your shared expenses on an inadequate income while you are putting funds into savings after your portion of shared expenses?  Because in the end, as a married couple, there is some expectation that you have shared assets and communal property at some point, and a shared happy future as well. And it only gets more so when you add children.

I think I would be more concerned with establishing a new way of managing money and expenses, both personal and communal, so that there is awareness, transparency, and shared commitments from both of you, something that eliminates this kind of reality. Getting him to engage in a new, better, more efficient 'family-friendly' way of doing things would seem to me to be the better focus than getting him to want to do an individual budget. Just the two cents of a stranger on the Internet who has no idea what your life looks like.

The Chains of Habit Are Too Light To Be Felt Until They Are Too Heavy To Be Broken

Baby steps