fix: Message signature is not appending (#10855)

# Pull Request Template

## Description

**Issue:** The message signature wasn't being appended to new email
conversations when a target inbox was selected.

**Solution:** To address this, a reusable `handleSignatureSetup`
function was created to manage the signature logic. The same logic was
applied in both cases, when the inbox selection changed (using `watch`)
and during the initial load (using `mounted`).

Fixes
https://linear.app/chatwoot/issue/CW-4005/allow-to-activate-the-message-signature-for-new-email-conversations-by
https://github.com/chatwoot/chatwoot/issues/10836

## Type of change

- [x] Bug fix (non-breaking change which fixes an issue)

## How Has This Been Tested?

**Steps to reproduce**:
https://github.com/chatwoot/chatwoot/issues/10836#issuecomment-2637354304

### Loom video

**Before**

https://www.loom.com/share/ccf597cfa8d94d0eaff1222102901d2c?sid=abfea42b-425e-446e-8e92-99359b786607

**After**

https://www.loom.com/share/d9deddfcf8de48ab87e31911dfb774d8?sid=c1aac19b-b243-428e-9a9f-2ad9f4efe49c


## Checklist:

- [x] My code follows the style guidelines of this project
- [x] I have performed a self-review of my code
- [x] I have commented on my code, particularly in hard-to-understand
areas
- [ ] I have made corresponding changes to the documentation
- [x] My changes generate no new warnings
- [ ] I have added tests that prove my fix is effective or that my
feature works
- [x] New and existing unit tests pass locally with my changes
- [ ] Any dependent changes have been merged and published in downstream
modules

app/javascript/dashboard/components-next/NewConversation/components/ActionButtons.vue

@@ -77,7 +77,7 @@ const toggleMessageSignature = () => {
   setSignature();
 };
 
-// Added this watch to dynamically set signature.
+// Added this watch to dynamically set signature on target inbox change.
 // Only targetInbox has value and is Advance Editor(used by isEmailOrWebWidgetInbox)
 // Set the signature only if the inbox based flag is true
 watch(
@@ -86,7 +86,8 @@ watch(
     nextTick(() => {
       if (newValue && props.isEmailOrWebWidgetInbox) setSignature();
     });
-  }
+  },
+  { immediate: true }
 );
 
 const onClickInsertEmoji = emoji => {