I found a weird behavior when I delete the private command from the package and when I first run a Quit command.

BeginPackage["TFPackageSourceCode`"];
(*WD1Int::usage="";
Begin["`Private`"];*)
WD1Int[f_,xR_,xL_,np_,nw_]:=Module[{xLcheck,xRcheck},xLcheck=NumberQ[xL];
xRcheck=NumberQ[xR];
If[xLcheck==True&&xRcheck==True,ResultTF=SetPrecision[NIntegrate[f,{x,xL,xR},WorkingPrecision->Re[IntegerPart[nw]]],Re[IntegerPart[np]]],"Enter numeric inputs for Lower and Upper Limits"]];
SetAttributes[WD1Int,{ReadProtected,Protected,Locked}];
(*End[];*)
EndPackage[];

and with the same main file.

Manipulate[
 DynamicModule[{f = Sin[x], xR = 1, xL = 0, 
   Res = 0.4596976941318603`16}, 
  Column[{Style["1D Definite Integral Calculator", "Function"], 
    Row[{"Lower Limit ", InputField[Dynamic[xL], Number]}, Spacer[5]],
     Row[{"Upper Limit ", InputField[Dynamic[xR], Number]}, 
     Spacer[5]], 
    Row[{"Function  ", InputField[Dynamic[f]]}, Spacer[20]], 
    Button[Style["Calculate", Green, Bold], 
     Dynamic[Res = TFPackageSourceCode`WD1Int[f, xR, xL, np, nw];]], 
    Row[{"Result", InputField[Dynamic[Res], Enabled -> False]}, 
     Spacer[51]], 
    Row[{"Plot the Function ", Checkbox[Dynamic[fp], {False, True}]}],
     If[fp == True, 
     InputField[
      Dynamic[Plot[f, {x, xL, xR}, Dynamic[PlotLabel -> f], 
        ImageSize -> Full]], FieldSize -> {30, 15}]], 
    Button[Mouseover[Style["http://www.wolfram.com", "Hyperlink"], 
      Style["http://www.wolfram.com", "HyperlinkActive"]], 
     NotebookLocate[{URL["http://www.wolfram.com"], None}], 
     Appearance -> None]}]], {{np, 16, "NumberFormat"}, 
  ControlPlacement -> Bottom}, {{nw, 16, "WorkingPrecision"}, 
  ControlPlacement -> Bottom}, 
 Initialization :> (Get[
     FileNameJoin[{NotebookDirectory[EvaluationNotebook[]], 
       "TFPackageSourceCode.wl"}]];)]

Thank you for the reply but to be honest I still did not understand why I am the problem. From the main code Sin[x] or let me say f[x] goes as a symbolic variable to the package. There I create a variable x and it is privative because the package is private. I thought creating an x variable in the module should have solved the problem.

BeginPackage["TF1DINTPackage`"];
D1Int::usage="";
Begin["`Private`"];
D1Int[f_,xR_,xL_,np_,nw_]:=Module[{xLcheck,xRcheck,x},xLcheck=NumberQ[xL];
xRcheck=NumberQ[xR];
If[xLcheck==True&&xRcheck==True,ResultTF=SetPrecision[NIntegrate[f,{x,xL,xR},WorkingPrecision->Re[IntegerPart[nw]]],Re[IntegerPart[np]]],"Enter numeric inputs for Lower and Upper Limits"]];
SetAttributes[D1Int,{ReadProtected,Protected,Locked}];
End[];
EndPackage[];

but it did not. And I did not understand your solution idea.