Adverbs within dialogue are fine, if real people use them (like really). I would avoid stuffing an adverb into dialogue, or using dialogue as a cheat to express with an adverb what should be exposition. The line "That horse nickered nervously," does not sound like something a person would say. They would more likely say, "She's nervous." or if you want to give them some expertise, "Look at her, she's clearly nervous,". (got an adverb in there!)
One of the problems with adverbs is they don't always point at an observer.
If I wrote The man clearly wanted to say more, on rewrite I would ask myself: Clear to whom? Bob? So I might rewrite, The man broke eye contact, and it was clear to Bob he wanted to say more.
All the adverbs have a adjective you can use instead. sad, nervous, clear, quick, rapid, loud, meek. When you edit for adverbs, remember they are shortcuts and typically used in speech because they are shortcuts. Find the adjective and write a sentence or phrase with it, instead. Or find the synonyms of it, and use one to describe it.
For "My father died," she said sadly I disagree that 'sadly' is not necessary. IRL I loved my father, but when I describe his death you will not detect sadness or anger in my voice, I can say "My father died thirty years ago," in a matter-of-fact tone.
Your context may be different, maybe her father just died yesterday, but even so describing the girl's demeanor and expression is an opportunity here, thrown away by the shortcut 'sadly' which is very ambiguous and non-impactful. You could describe her:
"John knocked and waited, smiling to himself about the joke the cabbie had just told him. Alice answered the door. Her eyes were red and puffy, bloodshot, her shoulders slumped. She had tissues balled up in her hand.
His cheerfulness vanished, replaced by concern. "Are you ill? Have you been crying?"
"My brother called me, and ... my father died," she said, and the grief in her voice almost broke it. The words triggered new tears. He stepped toward her to hold her, and she fell into him, sobbing.
Of course that might not fit your particular scene, but it is an example of how a lot of words can replace one adverb. When you edit for them, consider them an opportunity to have some fun, you don't have to replace them with just a few words or some grammatical rearrangement. You can replace them with whole paragraphs, and often you should do that.
I would generalize that observation as an aside: one of the worst misconceptions I see in new writers is this urge toward "efficiency" in writing. It isn't just the 'show don't tell' mantra, they also leave out what should be in there. That is the problem with monolithic blocks of back and forth dialogue, with nothing in-between or breaking it up, no internal thoughts, no silences, no actions of what people are doing or seeing or feeling as they speak.
In the interest of "efficiency" and getting through something quickly, they leave scenes under-imagined, meaning the reader is not seeing, hearing and feeling what you as the author intended, an under-imagined scene means you have not conveyed accurately what the characters are feeling and the changes they are going through. "Under-imagined" is a state the reader is left in, and sometimes because the writer failed to imagine the scene fully and fell into just dumping something on the page (like the text of a conversation). In that sense, for an author, adverbs should signal a place where they were 'telling' instead of 'showing'. e.g. telling me Alice was sad, instead of showing me through her actions and appearance that she is sad.
Readers don't care how many words they have to read, or if it takes you a whole page to get from John's remark to Alice's reply to that remark (I've seen that in published fiction). They just don't want to be bored by the writing, so what you write cannot be repetitive (or nearly repetitive; don't say the same thing using three synonyms for 'sad').
I know what I am about to say is fuzzy, but you want to convey the full scene using as few words as possible, but not fewer. So you want to discard irrelevant details (we don't care what morning show is playing muted on Alice's TV, or whether she is wearing shoes or barefoot, etc). If you can write something in fewer words by using a better adjective or metaphor or rearranging the sentence or breaking it up, do that.
But don't short-change the scene to save words. Try to keep in mind what is important and what you are trying to convey about the sensory scene (visual, auditory, olfactory, etc) and the emotional scene and mental context (relevant memories, thoughts, worries, priorities), and the physicality involved that reflects it (expressions, postures, interactive movement like for fighting, play, emotional or romantic contact).
You are 'efficient' when you can convey all that with as few words as possible, and cutting would mean sacrificing something important to the scene. The number of words depends on the scene, how important it is, and how complex it is. A carnival may take pages, or a standard hotel room just a short paragraph. Especially if the hotel room setting plays no significant part in the rest of the story, in which case it may not be described at all:
It had a bed, John dropped his luggage at the foot of it and fell into it. He woke about three hours later, found the bathroom and shower, then returned to the bed, set the alarm and slept another nine hours.
Most readers don't require a description of a hotel room that does not matter, and if John is exhausted by traveling for 36 hours or so, he would not be that observant anyway. What I am trying to say is, you have to pick what matters [makes a difference to the characters or events in the story], and find a way to describe everything that matters, but only what matters.
Bringing it back to adverbs: They are almost always shortcuts for telling instead of showing, thus they signal that descriptions that matter have probably been omitted, making the writing weak and under-imagined.