Replace every Minus to Plus in Expression

You have found the snags and you're right -- it's a simple matter. You just need the right rules.

For a pure symbolic expression you can use Kuba's suggestion.

a^2 - b + c*d /. -1 -> 1

a^2 + b + c d

For dealing with complex numbers you can use

(1 - b I) a /. x_Complex /; Im[x] < 0 -> Conjugate[x]

(1 + b I) a

If your expressions are more complicated than these, you might need more elaborate rules. But I offer more without knowing what form the more complicated expression take.

For most of these cases, judicious use of the undocumented function Internal`SyntacticNegativeQ[] works nicely:

Replace[{a^2 - b + c d, a - b, -a + b - c}, x_?Internal`SyntacticNegativeQ :> -x, {-1}]
   {a^2 + b + c d, a + b, a + b + c}

It will have trouble with complex numbers, however:

(3 - 4 I) (a - b I) E^(I t) /. x_?Internal`SyntacticNegativeQ :> -x
   (-3 + 4 I) (a - I b) E^(I t)

so just use Conjugate:

(3 - 4 I) (a - b I) E^(I t) /. x_Complex /; Negative[Im[x]] :> Conjugate[x]
   (3 + 4 I) (a + b I) E^(I t)

However, none of these can deal with -3 + 4 I; thus, a separate rule for real parts is necessary:

Replace[-3 + 4 I, {x_Complex /; Negative[Re[x]] :> -I Conjugate[I x]}, {-1}]
   3 + 4 I

Fold[Replace[#, #2, {-1}] &, (-3 - 4 I) (a - b I) Exp[-I t],
     {x_Complex /; Negative[Re[x]] :> -I Conjugate[I x], 
      x_Complex /; Negative[Im[x]] :> Conjugate[x], 
      x_?Internal`SyntacticNegativeQ :> -x}]
   (3 + 4 I) (a + I b) E^(I t)