Actually, it could be said two ways - plural and singular. There is no hard and fast rule.
"I'm trying to overcome my feeling of dread"
"I'm trying to overcome my feelings of dread"
Usually, if you add "of " after it, it is singular (feeling).
If you do not qualify 'feeling', as with 'feeling of excitement', you would probably use the plural.
"I'm trying to overcome my feelings of dread"
If you have feelings 'for' someone, it is usually expressed in the plural.
However, people would understand you whichever one you use.