They are very similar. The second one is just a bit more precise. Sometimes I use the second one to give the other person a more clear idea of what I'm feeling, especially if they are not an emotionally intense person.

Pelin... the first sentence is not said in standard good English. Even saying "I'm just feeling".... has no real sense out of some specific context. FeelingS is the noun to have feelingS. to feel is the verb. As in I AM just feelING happy. Used as a verb gerund form. The two sentence are not the same, feelings is one word and emotions is another word, and with the richness of English these two words are different Of the two sentences... your second is a correct and a 'said' sentence. I am experiencing intense emotions. For a similar meaning you could say, I am having many feelingS about 'this' - whatever this is. BUT again, many feelings and intense emotions DO not mean the same thing.

You could also say "I'm [emotionally] overwhelmed." The "emotionally" part can be omitted depending on the context.

As Collin says, they're similar. The first is more informal, and sounds like 'spoken' English. The second is more precise and academic, written English.