[C#] fix: Typing indicator is "stuck" (#2250)

## Linked issues

closes: #minor
## Details
Fixes the issue where the typing timer is stuck after the bot sends the
message.


#### Change details
- The send "typing" activity method is awaited to complete before the
send "message" activity is processed.

## Attestation Checklist

- [x] My code follows the style guidelines of this project

- I have checked for/fixed spelling, linting, and other errors
- I have commented my code for clarity
- I have made corresponding changes to the documentation (updating the
doc strings in the code is sufficient)
- My changes generate no new warnings
- I have added tests that validates my changes, and provides sufficient
test coverage. I have tested with:
  - Local testing
  - E2E testing in Teams
- New and existing unit tests pass locally with my changes

dotnet/packages/Microsoft.TeamsAI/Microsoft.TeamsAI/TypingTimer.cs

@@ -1,6 +1,7 @@
 using Microsoft.Teams.AI.Utilities;
 using Microsoft.Bot.Schema;
 using Microsoft.Bot.Builder;
+using Microsoft.Identity.Client;
 
 namespace Microsoft.Teams.AI
 {
@@ -20,6 +21,11 @@ internal class TypingTimer : IDisposable
         /// </summary>
         private bool _disposedValue = false;
 
+        /// <summary>
+        /// The send "typing" activity task
+        /// </summary>
+        private Task _lastSend = Task.CompletedTask;
+
         /// <summary>
         /// Constructs a new instance of the <see cref="TypingTimer"/> class.
         /// </summary>
@@ -100,7 +106,8 @@ private async void SendTypingActivity(object state)
 
             try
             {
-                await turnContext.SendActivityAsync(new Activity { Type = ActivityTypes.Typing });
+                _lastSend = turnContext.SendActivityAsync(new Activity { Type = ActivityTypes.Typing });
+                await _lastSend;
                 if (IsRunning())
                 {
                     _timer?.Change(_interval, Timeout.Infinite);
@@ -115,21 +122,22 @@ private async void SendTypingActivity(object state)
             }
         }
 
-        private Task<ResourceResponse[]> StopTimerWhenSendMessageActivityHandlerAsync(ITurnContext turnContext, List<Activity> activities, Func<Task<ResourceResponse[]>> next)
+        private async Task<ResourceResponse[]> StopTimerWhenSendMessageActivityHandlerAsync(ITurnContext turnContext, List<Activity> activities, Func<Task<ResourceResponse[]>> next)
         {
             if (_timer != null)
             {
                 foreach (Activity activity in activities)
                 {
                     if (activity.Type == ActivityTypes.Message)
                     {
+                        await _lastSend;
                         Dispose();
                         break;
                     }
                 }
             }
 
-            return next();
+            return await next();
         }
     }
 }