For avocados, I use the same methods as Kristin. Leave on counter until they feel very slightly soft. Then, put in fridge. If not eating entire avocado in one go, seal the un-eaten part in air-tight wrapping. It keeps for at least one day, although you may want to scrape off the brown layer.
In fact my diet is very similar to hers. Avocados, nuts, olives and salads form a large part of it. The exception is the evening meal when there is one "main" protein/fat dish (fish or meat) with low-carb vegetables.
I have been underweight (or close to it) for much of my life. The only exception was in the years prior to T2 diagnosis when I had put on weight. In the first few weeks after going on the low-carb diet I was ravenously hungry, but with time, that went away completely.
Early days. I'm only nine months into this diabetes thing.