Julie, about the job. You can't just agree to take on a second job. You might find a second job that works out great for you, you might not. You can't be coerced into finding a job. But you can be open to working with H towards common financial goals. I like Painter's advice of talking with a financial planner. Validate that you are aware that finances are an issue. Validate that you are open to exploring this idea further. But do not commit to anything.
I waited a whole year after BD to start my part-time job. For my own reasons, the timing was not right before now. (d in the hospital for 3 months, I wanted the summer with my kids, didn't see many part time jobs available, etc.) While I was interviewing for this position I received a call for an interview for a full-time position. H was excited about that and I told him I would talk to them but ultimately I would not accept a position that didn't allow me the flexibility to meet the needs of my children and my own needs at this point in time. I said I would be open to a more demanding schedule in the future, possibly the fall of 2016, possibly later. I could tell he didn't agree with me, but I am not negotiating on that. (this is the man who didn't understand why I didn't pursue a job that required 25% travel, while we have 3 children, AND he is away for 6 months. What does he think I would do with the kids??)
I believe I stated my intentions/goals in a confident and non emotional, non confrontational manner. He backed down. I am not saying your H will back down, just what worked for me. (also the recruiter for the other position fell off the face of the earth, so its not like it was a valid option.) I also started reading a book on financial planning and H seems to appreciate that. I do care about our finances, and I thought that was a small way to demonstrate that.
I think if your H is reassured that you care about the family finances and you are open to exploring ways to reach your financial goals, including possibly working more, that can go a long way towards validating his concerns. JMO.