No.
Structurally, this simply cannot work. All of the configuration (including fields displayed to the user) for any issue is determined by the project and issue type. If you try to change project by entering data in another field, JIRA will wipe out all the other fields the user has entered.
The best case is that you could do it, but you'll need some complex javascript that can save entered data and re-apply it as the screen changes, and it would have to be aware of every possible change of project and issue type.
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