Basically, this sentence is equivalent to saying:

The woman was lonely.

This is equivalent to:

The woman was tall.

So, the way "tall" is an adjective, "lonely" is also an adjective, depicting an attribute of "the woman". Hence, "lonely" is an adjective.

For example, you would never say:

The woman walked lonely.

The above sentence is describing the verb "walked" and so, we have to use an actual adverb. So, the correct sentence will be:

The woman walked lonelily.

Basically, "felt/feel" is a "linking verb" and not an "action verb" and hence, just describes the "subject/noun" and not the "verb".
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Hi HaSmamit,
I agree with EducationAisle. An adjective can appear after linking verbs, such as feel, appear, seem, and look. Some example sentences are:

I felt sleepy.
She seems upset.
You look exhausted.

Best Wishes!

'lonely' is describing a certain quality or attribute of the woman so 'lonely' serves as the adjective here.
The lovely woman ate happily - your logic holds true in this kind of sentence.

Question regarding adverb / adjective

Hope this provides some further insight.

How to get better at GMAT Sentence Correction. Be aware how adjectives and adverbs are different.
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