How can my husband obtain permanent residency in the U.S without having to leave the country?

To be honest, you aren't likely to get better advice here than you would get from two face-to-face meeting with actual immigration lawyers who have had the chance to evaluate your case. I would trust what they told you more than anything written here. I do think it's more than likely that they are right. If a person has not entered the U.S. legally ("with inspection"), even marriage to a citizen won't let them get a green card here without leaving. If they have accrued one full year of "unlawful presence" in the U.S., the act of leaving the U.S. - even if it is to process for a green card abroad - subjects them to a ten-year bar to returning. There is a waiver available for this bar based on hardship to a citizen spouse or parent (he wouldn't need to tell them why he he thinks he should be let back in, but why keeping him out would be hard on YOU), but these aren't always easy to get. There is another way even someone who entered illegally can get a green card here: if they had some type of green card case on file before April 30, 2001 that was approvable when it was filed, they are eligible for an old law that just let him pay a fine. But these days, few people are still around who are eligible for this. There may be some hope if the laws change with major immigration reform, but we would need to wait and see.