Both are correct, though the first sentence definitely is more common (or "the first sentence is definitely more common" either way is fine). The latter is perhaps more formal, and more suited for written language than it is for use in conversation.
In some cases there might be a different connotation, though:

I was suddenly feeling quite sick
I suddenly was feeling quite sick
I was feeling quite sick, all of a sudden (not really relevant here)

The first example stresses the sudden onset of sickness, whereas the second example puts the adverb first, to emphasize that it all happened quite, well, suddenly.
The third example stresses the feeling even more, and adds the fact that it happened suddenly in a sort of "in retrospect" kind of way.

Update:
As pointed out to me by Araucaria in the comments, my initial examples didn't quite work. So I'll have another go: Yes, both sentences are grammatically correct. The main difference is how "suddenly" is being used. In your first sentence, it's used as an adverb of manner. This means it modifies the verb (feeling):

I nodded, my chest gradually feeling heavier
As opposed to:
I nodded, my chest suddenly feeling heavier

It denotes the sudden nature of the action (feeling). The subject is either slowly becoming aware of a feeling, or is acutely aware of it. Your second example uses the same word as an adjective, modifying the comparative heavier:

I nodded, my chest feeling gradually heavier
I nodded, my chest feeling suddenly heavier

In this case, both sentences express the same action: the subject feeling something. The adjective modifies the comparative. It's not the speed at which the subject is aware of the feeling, but rather the speed at which the feeling manifests itself that changes.

Again, both are 100% correct, but the first sentence is more common. The latter less so, because in normal conversation, it's less likely for someone to describe a change in sensory awareness in detail. You will, however, find sentences like this in books quite often.

Original (incorrect) answer:

Both are correct, though the first sentence definitely is more common (or "the first sentence is definitely more common" either way is fine). The latter is perhaps more formal, and more suited for written language than it is for use in conversation.
In some cases there might be a different connotation, though:

I was suddenly feeling quite sick
I suddenly was feeling quite sick
I was feeling quite sick, all of a sudden (not really relevant here)

The first example stresses the sudden onset of sickness, whereas the second example puts the adverb first, to emphasize that it all happened quite, well, suddenly.
The third example stresses the feeling even more, and adds the fact that it happened suddenly in a sort of "in retrospect" kind of way.