I'm sorry that you're having trouble exporting your fonts. Thank you for sending your output and comparison output from InDesign.
Can you describe what you mean by the fonts are messy and share an example of a page where you notice the issue?
In the meantime, I would mention that setting Text Rendering to HTML text with Local Font Embedding (when possible)* will only embed fonts if the license you have for each font allows embedding.
If you don't have a license for a font that you'd like to use in your output, then set Text Rendering to HTML with Web-Safe Fallback Fonts and add web fonts from either Adobe Fonts (formerly Typekit) or Google Fonts. Here's how:
