I think companies are prepared for new hires to walk out the door. If you work in a field that's very close-knit and it might start rumors or get you a reputation, that's a little different. But my husband had a job in WI right after he got out of the Marines and it was only barely related to his training and paid like $10 or $12 an hour; he had applied for a different job in NY before he got the WI job and finally heard back from them and was hired through phone interviews, for something much more related to his career and at $24/hr. He was only at the WI job a month or so before giving notice, only gave 2 weeks (they needed him ASAP in NY), and his supervisors absolutely understood the situation.
My state (CT) has a 90 day 'trial employment' type period and I think that's pretty common in most states. In my book, if the employer isn't on the hook for unemployment/benefits/etc. before 90 days, the employee isn't on the hook for a high amount of company loyalty either. It's a way for both sides of the transaction to test the waters and see if it's a good fit.
But your husband should definitely try to get an offer in writing for his dream job -- if he had that, I don't know that I would keep applying for or start a different new job.
Edit: I don't know if this is possible in his field but it just occurred to me -- could he work with a temp agency? Even if it's only marginally related to his field, that might be a good solution for both him and an employer (and your bank account!).