Être vs avoir when using adjectives
Why is it that some adjectives use être (E.g. je suis desolé) and others use avoir? (E.g. j'ai peur)
Peur is a noun (fear) - as are faim (hunger), soif (thirst), chaud (~hot), froid (cold)...
As to why French people sometimes describe their feelings and states of being using nouns rather than adjectives - that is just the way things are after thousands of years of history shaping their language.
They think our ways of saying things are equally awkward when learning English. That's life.
To be clear, chaud and froid are adjectives but used with 'avoir', they behave as nouns. But the meaning will change. Il a chaud = he is (too) hot. Il est chaud = he is keen or he is horny. Il a froid = he is cold. Il est froid = He is dead or (when playing cache-cache) you're not near it.
Avoir and être are the verbs (to have and to be).
The other words in your examples are nouns.
"désolé" is not an adjective, it's the past participle of "désoler". La situation actuelle me désole (the current situation is saddening me) -> je suis désolé par la situation actuelle (I am saddened by the current situation). Here 'être' is not the main verb. It is the auxilary verb of the main verb 'désoler' in the passive voice.