In trying to understand your question, it seems you have already answered it. Therefore, I will help you support it.
Before the story, George hooked up with Lennie somehow. Most people don't go out of their way to choose a disabled person to befriend, however George seems to be rather welcoming of Lennie's companionship even if it is a pain most of the time. This was desperate.
During the story, we see several examples of lonliness leading to desperation. Curley's wife says she could have been in pictures. She seems to have struggled with her mom because mom hindered that life for her. I think running away and marrying Curley of all men (and it sounds like this happened on a whim) was a desperate act borne of the lonliness she felt being cooped up by her mother.
Crooks also took a shot at Lennie when Lennie came to visit with him as a result of lonliness. If Crooks had any practice at being friends, he would not have uttered the hurtful words he did to Lennie in suggesting that George would desert him.