Clear TUI input when starting new session

[OnlyYu1996]

Summary

• clear the TUI input widget when AppState::new_session() starts a fresh session
• extend the new-session action test to cover stale draft text being removed

Fixes PlatformNetwork/bounty-challenge#43779

Tests

• rustfmt --edition 2024 --check src/cortex-tui/src/app/methods.rs src/cortex-tui/src/runner/handlers/tests.rs
• cargo test --package cortex-tui runner::handlers::tests::test_handle_new_session -- --nocapture
• cargo test --package cortex-tui runner::handlers::tests -- --nocapture
• cargo check --package cortex-tui --lib
• git diff --check

Note: cargo fmt --package cortex-tui --check is blocked by an existing unrelated formatting diff in src/cortex-tui/src/question.rs, so I checked the touched files directly with rustfmt.

[greptile-apps]

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