Hi, in a word- Maybe. If I was putting a probable on it, I would say it is likely MS. I have go through temperature swings quite often. Your husband may have found his lower temperature tolerance. For me, depending on wind chill, I have problems below 52 degrees Fahrenheit. Below that temp, wear a pair of thinsulate gloves.
Although it may seem contrary, when I get a cold sensation in the summer, I immerse my hands or feet in cold water.
Skin sensitivity is part of the problem too. I do not use any spray on deodorant, but you may try a solid or a roll-on. He might also have sensitivity to the laundry detergent or fabric softener. I changed mine in 2003 after rashes started bugging me.
Your husband's neuro pathways have misguided or misfiring messages. This "short circuit" can manifest in many ways. The body's thermostat uses the pathways. Definitely see a neurologist to make sure what is MS and what is not MS.