No, "lonely" is not a verb. It is an adjective used to describe the feeling of being alone or feeling isolated.
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Both are correct, but they have slightly different connotations. "Feeling lonely" implies a sense of emotional isolation, while "feeling alone" suggests a physical state of being by oneself.
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Bee Gees are you feeling alright on these lonely, lonely
nights
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If you feel lonely you should take this time to spend with
friends and family!
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Feeling lonely is like being stranded in a desert with no one around.
Feeling scared is like being lost in a dark, labyrinthine maze.
Feeling lonely is like being adrift in a vast ocean with no land in sight.
Feeling scared is like being chased by an invisible threat in a dense fog.