It is possible that there is an infection causing the "bad odor". Usually when cells collect in the sinuses or nose, this causes the odor. If your husband will not try nasal washes, he may be resistant to using a nasal steroid spray, which is another approach to opening the airways. If this problem persists, you may want to see an ENT specialist for another opinion.
I have had the same problem for over a year and I think I am finally beginning to realize the problem. Nasal dryness! I have been drinking 8-12 glasses of water a day-no coffee or caffeine at all. I have alos been rinsing my nose with salt water to add moisture. It's helped alot its reduced the smell about 80% yet I still have a pretty dry mouth as well which resulted from the nasal dryness. I still have a bad smell when I speak because my mouth was so dry for over a year. I have an apt with my dentist next week to try and get a prescription to help increase my saliva.
I hope this helps. For me, I went to ENT after ENT thinking it was post-nasal drip which caused the problem--It was nasal dryness all along, they put me on nasal steriod which made the problem worse (all of this over a year). I finally stopped all the meds, did research on the net and realized it was my nose and mouth all along, yet they were dry.
I also bought a humidifier it helps at night. I sleep well without waking up with because of congestion. I hope this helps your husband--I know first had the embarrassment something like this causes and the stress since you can't find an adequate solution.