fix: fix search input stylings

[amareshsm]

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• [x] I have read the contributing guidelines.

What is the purpose of this pull request?

What changes did you make? (Give an overview)

Changes made:

1. Fixed the search text overlapping with the close icon
2. Correct the position of the close icon
3. Adjusted the positioning of the search results dropdown

Before	After

similar to https://github.com/eslint/eslint/pull/18682

Related Issues

Is there anything you'd like reviewers to focus on?

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