"I kept searching" is a correct and usable part of a sentence in written English.
For example, you could use it in a sentence such as: "I kept searching for a solution, but I couldn't find one."
"But my husband and I kept searching for help.
Still, I kept searching for a simpler one, one without pockets or a ruffled trim.
"I kept searching for any evidence that can justify Israel's action but didn't find any," Mr. Zaanin said.
Every time I went back for a fitting (there were so many fittings!), I kept searching for that Big Sky feeling.
For the next 2 years, I kept searching for my true passion.
I kept searching, and found that there is a fervent and passionate following for the Be-Ro Cookbook.
It intoxicated me and I keep searching for that feeling.
"I keep searching for some clue that will explain why it all happened," Ms. Rodríguez said.
Meanwhile I keep searching for the "numinous feelings" and "bizarrerie de l'ambiance" promised by shrooms.
I keep searching the crowded room, but there is absolutely no one from my class, and I'm disappointed.
Armed with knowledge from Norbert, I keep searching, setting my alarm and taking a long-cut across Tempelhofer on my way to work, or going at dusk to a square near my flat in Kreuzberg where he told me there was a nesting site.
