The phrase "alternately trying" is correct and usable in written English.
It can be used to describe a situation where two or more actions are being attempted in succession or in a back-and-forth manner.
Example: "She was alternately trying to solve the puzzle and asking for help from her friends."
Alternatives: "taking turns attempting" or "switching between efforts".
He was rewinding the tape, pressing the "stop" and the "play" buttons alternately, trying to find the beginning of the song.
Since then, KFC has careered from one campaign to another, alternately trying to revive Sanders or ignore him; the reprises have ranged from animated cartoons to archival film to actors impersonating his looks or his voice.
He left the impression that either he had been running for President all along or that he was still running for senator, or that his unlucky fate as a nationally viable Democratic candidate from a stubbornly red state would condemn him to a Sisyphean struggle of alternately trying for both for the rest of his life.
Tradition calls for the bull to be stabbed to death as he staggers around the field alternately trying to charge at, and escape from, his bemused predators.
In his book "So Far, So Funny, Hal Kanterrecounteded the year he spent producing "Kraft Music Hall," alternately trying to tame the meddling, egomaniacal Berle and giving in to his demands.
Foyer's Mick, a believable mixture of guilt and eagerness to be gone, offers sheepish justifications for his decision to sell Tommy's house out from under him, while alternately trying to elicit sympathy for the necessity of the sale and attempting to deflect blame with lashings of heedless cruelty.
In this Old World, the United States finds no comfort and alternately tries to withdraw from it and reform it.
American diplomats have alternately tried to charm and chasten Mr. Saakashvili, opposition politicians, and News Corporation and Imedi executives into making peace and toning down their disagreements.
The kids appealed to me with loud wails, "Ms. Mooozzzlleeee, nigger put toothpaste in my hair!" or alternately tried to "get away" with things.
(Alternately, try to move-in with a roommate who's mid-lease). This way, the person on the lease is the one with more solid credit.
He had alternately tried being more patient and more aggressive.
