Once I leave Mail Compose Screen on iOS, the unsaved message ceases to exist.

Drafts should get saved to the Drafts folder of the account selected when composing the email, if you tap on the Cancel button at the top left, and then tap on Save Draft rather than Delete Draft.

Otherwise, if you swipe it down, it will remain at the bottom of the screen, until you cancel it or actually send it. While at the bottom of the screen, it is not saved anywhere. You can bring it back up, by tapping on the title.

As to searches, the Mail App will search inside the email body, but it just does not offer a specific option for it. The default search will include the contents of emails. If you select one of the filter options presented such as Subject, Attachment or Sender then it won't include the body in the search.

To have it saved as a draft you have to tap on Close and then select to Save Draft. If you directly close the app, it might not save properly. As for other mail app you can use whatever you find best for you. Personally I have been using Mail for years and like it a lot.

db_1212 wrote:

update: turns out Apple ignores this app so hard that it doesn't even appear as a topic...anywhere when selecting a topic for a post in here.

There are lots of iOS apps that aren't listed under that category including Safari, Health, Music, TV, Weather, Stocks. I could go on but I think I've made my point.

They do have an entire support section on Mail:

Mail - Official Apple Support

But, if you prefer another mail client, you should, by all means, use one.