Feel bad or badly?

It’s ‘feel bad‘, plain and simple.

All sense verbs are followed by adjectives.
This beer tastes bad. She looks good. This coffee smells nice. My blanket feels soft. This tune sounds amazing. He seems interesting.

When we feel bad we are not happy. If we were to say that we feel badly, it would imply that something was wrong with our sense of touch.

I feel good. – I am happy.
I feel well. – I am healthy.

I hope that this explanation helps.