I want to ask question that would solicit more than one acceptable answer. Can I do this? It's quite embarrassing to just keep the question opened but if I accept an answer, that would stop the question from getting better answers over time.
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No one is going to stop you from hesitating to accept an answer. The choice is entirely yours, and you should wait until you get the answer that helps you the most. And if it never comes, it may or may not be a problem with the question (you would know if it was—downvotes and close votes are an indicator).
But in any case, I don't think it's necessarily possible for a question to warrant more than one accepted answer. There will almost always be more than one correct answer, but accepting is one-and-done; you pick the one that fits best in your situation.
And finally, to the last point: I definitely don't care if an answer has already been accepted unless my answer would have been identical. If I have useful information to add, I will post another answer, because the user can always accept a different answer. I'm sure many people do this.