"as I continued" is correct and usable in written English.
It is often used as a transitional phrase indicating that the speaker or writer is still moving forward in their narrative. For example: "As I continued down the winding road, I noticed the sun beginning to set."
Exact(60)
Things improved as I continued my search.
As I continued east out of Baie-St.
As I continued to perform Shorty, I didn't expect the spiralling level of interest.
So as I continued blowing the leaves I just thought and felt what I felt.
As I continued walking, I asked a local about these informal tobacconists.
As I continued through the maze, though, I became a little afraid.
But as I continued with my career, not only did I grow, but the sport grew.
I admit that, as I continued to read, I was nagged by cavils.
My voice would then quake as I continued, which only made it quake more.
But as I continued writing, my encounters with these strange contagions persisted.
"But as I continued to come, it became therapeutic and a comfort to me.
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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