I want to draw on traditional, arguably "cliché" (?) fantasy species, like elves, orcs, goblins, dwarves, faeries, etc.
How can I involve some of these older elements, while leaving behind the racist subtext some of them carry?
Racism is about Mental Traits.
The problem with racism is the assumption that mentality, morality and agency are inherited qualities. The problem is not that the shape of my nose or the color of my skin are heritable, it is the assumption that this means something (predicts something) unchangeable about my attitudes toward others, my intelligence or stupidity, my ability to make decisions for myself, how hard I am willing to work, whether my greed precludes sympathy or fairness, whether or not I am prone to violence or rape or thievery or deception for selfish gain, even whether or not I am able to clean myself.
Note that all the racist tropes are basically about externalities predicting mental traits. The same goes for other bigotries; assuming women are not as smart as men, cannot do math, or that a pretty woman is automatically short-changed in intelligence, or is less competent in a strategic or intellectual job than a male.
Simply swapping around skin colors (or genders) isn't enough to break the racist associations. In a movie that might be part of the solution, because they are on the screen every minute. In a novel, you can't keep repeating that your "goblins" are tall blond Norwegians with celestial noses. Eventually the reader is going to fall back on the stereotypical description they have been trained to hold, and ignore your description.
IMO, to break through the racism and bigotry, you need to break the connection between their physical appearance or morphology and what MLK called "The Content of Their Character".
But the thing you want to break is the assumed correlation, so it isn't enough to say goblins (with hooked noses) are all now accomplished musicians and live in communes and share all the money they earn. That is still maintaining a racial connection, it is still saying that their morphology predicts the content of their character. To break the correlation, goblins that all look similar must come in every variety of personality; i.e. their appearance does NOT predict anything about their personality or proclivities or intelligence or behavior.
The same goes for other characters; if you just swap around skin colors and nose shapes and hair types, you are still reinforcing the notion that the body you are born with determines the content of your character.
Unfortunately for your project, avoiding the stereotypical assumptions destroys the utility you would get by drawing on the traditional "elves, orcs, goblins, dwarves, faeries, etc."
Authors use those as a shortcut to signaling the content of their character, and describing the physical characters. When we hear orcs in the woods, we think "danger", we don't think of calling to them for help and them coming to help, give us a pat on the back and a hoagie for the road.
If you use them, you get them with their racist baggage. If you want to avoid the racism, you need to invent your own races, or just go raceless: Don't describe morphology, or when you do ensure it is not in the vein of "All redheads are irish, and all Irishmen are drunks". i.e. you may want a character to stand out by making them a redhead, but that doesn't make them "hot headed" or determinative of anything about them; and nobody in your world expects it to be.
For example, if Goblins really are individuals without a common personality, your characters would not assume anything about them; just like whites in America don't automatically assume they can trust other whites. Other whites can be serial killers, thieves, con-men, rapists, drug addicts, selfish greedy bastards, who knows what? Whites are represented in every category of human depravity.
In order to judge somebody by the content of their character, you have to get to know them first, and if you wish to portray non-racist characters then they cannot take the shortcuts of using morphology to predict character content.
I fear that destroys the utility of using the fantasy stereotyped races (quickly conveying character content), and it is part of the reason I never use them.