"eschewed being" is correct and can be used in written English.
It is a verb phrase meaning to deliberately avoid or abstain from something. It is typically used in a formal or literary context. Example: She eschewed being the center of attention and instead preferred to work behind the scenes.
Exact(2)
Despite all of Brazil's accomplishments in surging ahead of Venezuela in Latin America, Mr. da Silva has eschewed being labeled a leader in the region.
Until the 20th century, the Yucatán was more closely connected with Europe and the United States than with the rest of Mexico, and the people of the Yucatán (mainly Maya) have generally eschewed being called "Mexicans".
Similar(58)
Are people clamoring for Chloe because she eschews being on magazine covers (why bother when The New Yorker already had Jay McInerney profile her as " super-cool girl" six years ago)?
Despite her status as a member of technology's billionaire club, she eschews being ferried by private jet from her offices in Taipei to Silicon Valley.
To this end, words are eschewed, signs are limited.
Eschew is still current, but it isn't used much – perhaps it's too disdainful, too prideful.
Our sportswriting is often as elegant as anything we publish, but I was surprised to see that five of our last 10 "eschews" were in sports contexts, where football teams have been "eschewing" field goals and "eschewing" the safety of the running game.
A trailblazer on the training front, McKeever eschewed what was long the conventional wisdom that more mileage is better and tailored her workouts around technique.
The idea of what a voguing drag queen from Harlem would do if plunked down in an environment that eschewed presentation is amusing to contemplate.
Is it circularity in the specification of sets that is to be eschewed?
Dairy, alcohol and meat are out; fish is permissible, and salt is eschewed by many.
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Since I tried Ludwig back in 2017, I have been constantly using it in both editing and translation. Ever since, I suggest it to my translators at ProSciEditing.

Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com