It is from the verb 「通{とお}す」.

>「Verb in 連用形{れんようけい} (continuative form) + 通す」

means:

>"to continue [verb]-ing to the end"

In 「しどおし」, the first 「し」 is the 連用形 of the verb 「する」.  「どおし」is the 連用形 of 「通す」.  The と-to-ど change is [rendaku][1].

[1]:  https://en.wikipedia.org/wiki/Rendaku

Thus, 「どおし」 has nothing to do with「同士」 -- none.

「通す」 is written 「とおす」 in kana, not 「とうす」.

My TL of the line:

>"Even if they call me a natural worrier, I just can't stop feeling uneasy!"

Other examples:

「朝まで飲み通す」 = "to keep on drinking till the dawn"

「９回まで投げ通す」 = "to continue to pitch till the 9th inning"

「仕事をやり通す」 = "to completely finish one's work"