Which one is right to say?
- He felt embarrassed
- He felt embarrassment
Is there any difference between them?
Normally the first. The verb "feel" can have an adjective complement, and as we are describing someone's feelings, the adjective works well. Compare
He feels happy. (Normal)
He feels happiness (Rare)
The noun form might used rarely, for example:
John felt something he had not experienced for a long time, embarrassment, even guilt.
Here a noun is required, as it has to be "something..."