I'm totally wild guessing here:

Scarlet

Because:

It seems to me that by adjacent we should consider words that are "connected" via another word. That is, not the words x and x+1, but x-1 and x+1. So for example we pair "letter" and "number" and get "symbol"

So, we get a partial solution for some of the words:

scarlet
fruit - A (scarlet letter)
letter - apple
computer - symbol
number
subject - consumer
consume - history
before -
cut - happy
cry -
slide - weep
gentle -
awake - rought
not -
rain - hot
cold -

We basically need to find a color:

the color must fit with rain and letter. So I'm guessing scarlet for: "scarlet rain" (because of this or perhaps blood in general) and the The Scarlet Letter, which is the letter "A".

I'm totally wild guessing here:

Scarlet

Because:

It seems to me that by adjacent we should consider words that are "connected" via another word. That is, not the words x and x+1, but x-1 and x+1. So for example we pair "letter" and "number" and get "symbol"

So, we get a partial solution for some of the words:

scarlet
fruit - A (scarlet letter)
letter - apple
computer - symbol
number
subject - consumer
consume - history
before -
cut - happy
cry -
slide - weep
gentle -
awake - rough
not -
rain - hot
cold -

We basically need to find a color:

the color must fit with rain and letter. So I'm guessing scarlet for: "scarlet rain" (because of this or perhaps blood in general) and the The Scarlet Letter, which is the letter "A".