How can I prevent rubber handlebar grips from melting/disintegrating?
I remember this would happen to me as a kid while on my bmx bike especially in humid summer weather. 

So you are buying new grips already - I'd say make sure they are a harder compound than whatever your current ones are. Seems like harder rubber is less likely to deform/melt away. 

With your current grips you might try wrapping them in bar tape. There are many kinds. I use cork - is good for wicking away moisture from sweaty hands but it is thick. Synthetic bar tape that's thinner might wrap tighter and be less bulky around your grips. If you don't have any bar tape, maybe try the clear plastic wrap from your kitchen. It should keeps the little bits off your hands. Could be more slippery so be careful.