Is your hubby fully rinsing ALL of the soap off of baby? If baby is sitting in a little bath tub, and your hubby uses lots of soap, perhaps he's having a reaction to the unrinsed soap.
Do you use a lot of soap or fabric softener in your towels and bedding? Fabric softener can bother sensitive skin...and be hard to rinse out completely. Put your towels into the washer with no soap. If you see bubbles while it's agitating, all of the soap is not getting rinsed out. Many people use way more than is needed, thinking it will mean cleaner laundry. It just means that your laundry doesn't get fully rinsed.
You said that you've been feeding some new foods. If you've introduced some new fruits or veggies, you might be onto something. Chicken would be an odd one...not many are allegic.
The exersaucer might also be a culrit, thought I doubt it.. Instead of using cleaner on it, you might consider taking it outside and hosing it down, washing with Dawn, and then rinsing again.