Refactor SearchAppBar: enhance search functionality by adding search icon and listener for search text changes

lib/widgets/search_app_bar.dart

@@ -28,15 +28,19 @@ class SearchAppBar extends StatefulWidget {
 
 class _SearchAppBarState extends State<SearchAppBar> {
   late TextEditingController _searchController;
+  void _onSearchChanged() {
+    setState(() {});
+  }
 
   @override
   void initState() {
     super.initState();
-    _searchController = TextEditingController();
+    _searchController = TextEditingController()..addListener(_onSearchChanged);
   }
 
   @override
   void dispose() {
+    _searchController.removeListener(_onSearchChanged);
     _searchController.dispose();
     super.dispose();
   }
@@ -79,11 +83,6 @@ class _SearchAppBarState extends State<SearchAppBar> {
                           hintStyle: TextStyle(
                             color: Colors.white.withValues(alpha: 0.5),
                           ),
-                          prefixIcon: const Icon(
-                            Icons.search,
-                            color: Colors.white70,
-                            size: 18,
-                          ),
                           suffixIcon: _searchController.text.isNotEmpty
                               ? IconButton(
                                   icon: const Icon(
@@ -100,7 +99,14 @@ class _SearchAppBarState extends State<SearchAppBar> {
                                     minHeight: 36,
                                   ),
                                 )
-                              : null,
+                              : const Padding(
+                                  padding: EdgeInsets.only(right: 8),
+                                  child: Icon(
+                                    Icons.search,
+                                    color: Colors.white70,
+                                    size: 18,
+                                  ),
+                                ),
                           border: OutlineInputBorder(
                             borderRadius: BorderRadius.circular(8),
                             borderSide: BorderSide(