'lo Mr. Stavland,
"in search of a better life" is correct English, "in searchy for a better life" is not. "searchy" may be used rarely in some special contexts, or may not be a correct English word.
Maybe I can tell you more if you post some context info.
"in ________ of" are used together with some words; "in [place, times, conditions] of", etc. "in ________ for" are used together with some words; "in[time, exchange]for", etc.
Would you like to find the grammar explanation for this and post it on this forum?
I tyvm for granting me the privilege and pleasure of answering your questions, and invite you to post again.