Migrate prompt caching tests to test_codex (#6605)

To hopefully fix the flakiness

codex-rs/core/tests/common/test_codex.rs

@@ -71,6 +71,7 @@ impl TestCodexBuilder {
         resume_from: Option<PathBuf>,
     ) -> anyhow::Result<TestCodex> {
         let (config, cwd) = self.prepare_config(server, &home).await?;
+
         let conversation_manager = ConversationManager::with_auth(CodexAuth::from_api_key("dummy"));
 
         let new_conversation = match resume_from {
@@ -79,15 +80,20 @@ impl TestCodexBuilder {
                     CodexAuth::from_api_key("dummy"),
                 );
                 conversation_manager
-                    .resume_conversation_from_rollout(config, path, auth_manager)
+                    .resume_conversation_from_rollout(config.clone(), path, auth_manager)
+                    .await?
+            }
+            None => {
+                conversation_manager
+                    .new_conversation(config.clone())
                     .await?
             }
-            None => conversation_manager.new_conversation(config).await?,
         };
 
         Ok(TestCodex {
             home,
             cwd,
+            config,
             codex: new_conversation.conversation,
             session_configured: new_conversation.session_configured,
         })
@@ -131,6 +137,7 @@ pub struct TestCodex {
     pub cwd: Arc<TempDir>,
     pub codex: Arc<CodexConversation>,
     pub session_configured: SessionConfiguredEvent,
+    pub config: Config,
 }
 
 impl TestCodex {