The display HAS to change because you are moving a paragraph where the cursor is sitting to another location but you don't need to move the cursor.
Code:
Sub Move_To_Next_Monday()
Dim oRng As Range, oSel As Range
Set oSel = Selection.Range.Paragraphs(1).Range
Set oRng = ActiveDocument.Range
oRng.Start = oSel.End
If oRng.Find.Execute("~Monday") Then
oRng.Start = oRng.Paragraphs(1).Range.End
oRng.FormattedText = oSel.FormattedText
oSel.Delete
End If
lbl_Exit:
Set oRng = Nothing
Set oSel = Nothing
End Sub