This sooo applies to me, not that i have had anyone singing 'Show me the way to go home' but the fact that balance issues can be conceived as being drunk.
I have MS and before the need to use a stick, i was very much aware that my staggering and swaying about left people looking and coming to the WRONG conclusion that i was drunk ( even at 9am ?!?!? )
Regarding feeling the need to explain to the neighbours, i would in a nice way.
It certainly is NOT their fault, no way would i speak to them in an offhand manner, i would simply explain the situation.
Obviously if they know the situation and still continue to insinuate he is drunk, then THAT is a different matter, but as it is of now, they are unaware of his health issues, so i would go round and see them yes, in a friendly manner.
I am sorry your husband is suffering in this way i hope soon it is treated.
All the very best to your husband.