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.
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"