How to convert input expression to FullForm String

You just put Unevaluated in the wrong spot:

SetAttributes[fullFormString, HoldAll];
fullFormString[expr_] := ToString[Unevaluated @ FullForm[expr]]

For your example:

fullFormString[Integrate[\[Alpha]^2, {\[Alpha], -3, 3}]]

"Integrate[Power[\[Alpha], 2], List[\[Alpha], -3, 3]]"

Maybe this way?

SetAttributes[fullFormString, HoldAll];
fullFormString[expr_] := ToString[HoldForm[FullForm[expr]], OutputForm]

Or maybe this way?

SetAttributes[fullFormString, HoldAll];
fullFormString[expr_] := 
  StringReplace[
    ToString[FullForm[Unevaluated[expr]], OutputForm], 
    "Unevaluated[" ~~ Longest[u__] ~~ "]" -> u]

fullFormString[Integrate[α^2, {α, -3, 3}]]

"Integrate[Power[\\[Alpha], 2], List[\\[Alpha], -3, 3]]"