You're looking at coming up with maybe $10,000 in order to refinance. That's not a massive amount of money.
Over the next year, your husband needs to keep shopping for a better-paying job (or better-paying part-time jobs) and you also need to be bringing in whatever income you can. Mow lawns, walk dogs, babysit a kid, rent out a room, whatever.
Also, look around the house and see if you have anything to sell. Do you have anything non-essential that you can cut? Cable, cell phones, food? With your hurdle being as low as $10,000, every bit will help.
If you guys can find a way to bring in $200 per week extra (or cut expenses), you could have this house refinanced to something reasonable within 12 months. Maybe that's not possible, but $100 a week makes it under two years to refinance - and that's not a horrifying timeline either.