FullForm with context for each symbol?

1. To not mess with $ContextPath you can Block it just for that evaluation. e.g. Block[{$ContextPath = {}}, ...].

   But we don't want to do that for the evaluation, input for FullForm should be evaluated freely and only the result should be returned with full symbols' names.

   To achieve that you should know:

2. $ContextPath and $Context affect the typesetting. MakeBoxes will strip those contexts from an evaluated FullForm.

So, to deal with 2 we have to perform MakeBoxes on our rules (local contexts values) and order that to not be touched by MakeBoxes anymore (RawBoxes):

Block[{$ContextPath = {}, $Context = "other`"}
   ,
   RawBoxes @ ToBoxes @ FullForm @ {x, 1}
] 

System`List[Global`x,1]

It is very convenient when we want to "toggle a visibility" of specific contexts:

Block[{$ContextPath = {"System`"}, ...

 List[Global`x,1]

Based on Kuba's idea, this may also work:

RawBoxes@MakeBoxes[FullForm[p + q]] /. 
   x_String :> Context[Evaluate@Symbol@x] <> Last@StringSplit[x, "`"] /; NameQ[x]