I'm really struggling with the distinction. In English, I often don't think that I make much of a distinction between "what" and "how" I feel.

¡Hola Brett! We can see that it can be difficult to distinguish the nuance between these two expressions. In Spanish, the first question of "WHAT do you feel for that guy?" could be translated into "¿QUÉ sientes por ese chico?" and the answers could be "siento amor, odio, etc. por él", and the second one of "HOW do you feel about that guy?" could be translated into "¿CÓMO te sientes con respecto a (regarding) ese chico?" and you could answer "Me siento bien, nerviosa, mal, etc." As you can see, we use nouns to reply to the first question, while we use adjectives or adverbs in the second one.

I hope that this helps.

Gracias, Silvia.