Optimization: list manipulation

ClearAll[f]
f = MapIndexed[If[LessEqual @@ Abs @ #,
  {Symbol["x" <> ToString[#2[[1]]]],  Sign @ Last @ #}, 
  {Sign @ First @ #,  Symbol["x" <> ToString[#2[[1]]]]}] &];

list = {{-5, 1}, {7, -3}, {4, 4}}; 

f @ list

{{-1, x1}, {1, x2}, {x3, 1}}

Also

symbolpositions = MapIndexed[Flatten[{#2, #}] &][2 - Boole[LessEqual @@@ Abs[list]]];

symbols = Symbol["x" <> ToString[#]] & /@ Range[Length@list];

ReplacePart[Sign@list, Thread[symbolpositions -> symbols]]

 {{-1, x1}, {1, x2}, {x3, 1}}

MapIndexed should be the suitable function working at the overall level. A version slightly different from that of @kglr 's, with OrderingBy, ToExpression, and StringTemplate

symbol = ToExpression[StringTemplate["x``"] @@ #] &;
func = If[Greater @@ OrderingBy[#, Abs],
    {Sign[#[[1]]], symbol[#2]}, 
    {symbol[#2], Sign[#[[2]]]}
] &;
list = {{-5, 1}, {7, -3}, {4, 4}};
MapIndexed[func][list]

{{-1, x1}, {1, x2}, {x3, 1}}